You need to cut him some slack. It sounds like he may have some additional needs and be masking at school.
This is exhausting for them and often means they lash out at home as they feel safe.
You need to seek more support for him so you can understand how to parent him effectively and stop beating yourself up over his behaviour.

As he has an EHCP I'm assuming he has a disability
Is he masking at school?

he's obviously masking heavily at school cut him some slack

It sounds to me like he's masking at school which is incredibly common.

So, gently, YABU.

I think you’re looking at it the wrong way. They expend huge amounts of energy being good at school. Then at home they let out all the tension to their safe person. Which is you. I don’t know what the answer is and I know it’s hard but constant punishment creating a large number of head-to-head battles between you will only make you both more exhausted. Does he have plenty of running around time after school to burn off pent up energy?

btw my children are being difficult too, this point in term is hard for them I think.

He may be masking at school
But even some NT kids do behave better at school. In those cases the reason imho is:

• absolute consistency in terms of expectations and consequences, including seeing these applied continually to peers
  

• lack of emotional levers to pull. An unrelated adult simply doesn't react in the same way to emotional responses and children can easily detect this
• experience & perspective: some teachers have been hundreds of kids, from a neutral perspective. They can draw on that bank of experience to find effective strategies more quickly than you

It sounds like he is basically doing the fizzy drink thing. Holding everything in while at school and it is all bubbling away thoughout the day and getting shook up inside of him. When he gets home - into his safe place - and the lid comes off and the fizzy (ie him) explodes.
He can't help it and he shouldn't be punished.
Make an appointment with the senco to find some ways to help him with this.

We had this with DS (suspected ASD, waiting for assessment), in his case behaving well and fitting in in school was a massive effort for him and the stress that it caused had to come out somewhere.

educated guess he has some sort of disability- he’s working very hard at school to behave as he “should”
he’s letting off steam at home where he feels (rightly) safe and loved
you need to find a way to help him decompress
think of having a very long day at work on meetings with people you don’t particularly like having to be polite, maybe listening to things that are not relevant to you, and when you’re done you’re mentally exhausted- you may go to the gym, or relax while reading or watching TV
now imagine getting home after a hard / stressful day and being told by your OH he invited people that you don’t like over and he asks if you could do a special dinner for them (assuming you’re a very good cook) - bet you will take it on your OH, not only inviting people you don’t like, but the stress of the day, now you cannot relax after a hard day
that’s what you’re asking him, behave all day, mentally exhausting and now do the same at home
of course it’s not a excuse to let him run wild, but you need to find his way of relaxing and recharging (be it watch a show on TV, run around in the park, nap) so he can cope with the social expectations

Look up the coke bottle effect. Then go back to the drawing board. He's little yet for high expectations. Help him find coping strategies.

What @Wnikat and @BillHadersLeftEye said.

It is SUCH hard work for him just to 'cope' at school, that it all comes out at home / with you when he relaxes. He knows he is loved unconditionally at home, so unfortunately family get the brunt of all the effort he is putting in at school.

Sounds like he's masking at school. What's his diagnosis? ADHD?

Another vote for coke bottle effect.

all that frustration of little shakes during the day in order to perform means when he is in his safe place the lid comes off.

does he have an area he can go to to relax and safely let out his frustrations?

You are looking at it the wrong way around. Have a watch of this (the next one will likely be automatically suggested, so you can continue with the talk) and then if you want to know more, go to the website www.livesinthebalance.com

I think everyone has said what I was thinking. My son has Asd is exactly the same. You need to manage your expectations and also realise they can’t mask forever . My sons slipped at around 9/10 when he just couldn’t cope anymore.

pre school he goes in, I’ll let him know if it’s me picking up or his dad/brother etc and what will happen after school. Knowing these things helps him and then he has time to play after school or release some of the tension so Lego/park/scooter no homework till later if at all depending how he is sometimes he’ll just not take anything in. It’s hard and it’s full on but it’s not you. Your son perhaps needs more support at school. Time out/movement breaks my sons school are pretty poor everything I suggested they say they can’t do or he won’t go. Because he doesn’t like change and he takes ages to warm up to new places and things. Understand his reasons for this but also make your boundaries clear, what you’ll accept re behaviour and what you won’t. My son swears when he gets worked up but it’s diminishing as he gets older and learns to manage things a bit better but he doesn’t hurt anyone or damage anything, you’ll find your own ways of dealing with it but not going to say there won’t be bad days and tough times but you’re not alone.

Another one suggesting it’s the coke bottle effect. Masking at school and exploding at home. It suggests there are unmet needs at school and his EHCP needs reviewing.

Like others said sounds like he's masking

Totally normal behaviour. My 7 year old is gifted and talented, the most academically able child in his year group apparently. I'm told all about his maturity at school.

He can be an absolute nightmare at home sometimes. It's their safe place. They have to behave and confirm all day, at it all comes out T home. You have my sympathy, it's not easy is it?

if Your child has an echp, someone should have explained that this is incredibly common. Children use up all of their energy to perform at school. When they get home to
their safe space and can finally relax they let out all those pent up feelings. It is also a sign that your child feels comfortable with you. Your child can be themselves

you can help by providing planner decompression. What exactly that means will vary by child. For mine, she needs quiet, privacy, and some mindless screen time. Another child will need to go to the park and run off energy.

Also as wrong as the pushing the other child obviously was why did he do it?

was he struggling with the hustle and bustle of end of the day? Where there too many people around and he just lashed out?

when you say you took the tablet away for xx when did that incident happen? If it wasn’t immediate he probably doesn’t link the two.

its hard work and it’s like your always trying to find a missing piece to the jigsaw but as someone many years down the line it does help to step back and realise that it’s not something personal, or something you are doing wrong it is just part of who he is and your role is to try to help him develop coping strategies

He is masking. Using his tablet might be the way he decompresses and self regulates, and you have just removed his coping strategy. You need to be softer, not harsher. As in ‘my child is having a hard time, not giving me a hard time’

Whilst you probably don't mean to and it will come from frustration you are being unfair on him. He will be masking at school, which is exhausting, and he just can't keep it up at home, he is only 6 and has additonal needs so he will really struggle until he gets into his routine and starts to mature.

I agree with others that it sounds like he’s masking and finding it exhausting so let’s it out when he’s not in school. We have the same here although it’s improving in some ways with age.
What has helped here is -
Emotion Coaching after something has happened.
Lots of opportunity to run around the park at few times a week after school.
Time to decompress when he gets home, with a drink and a snack and time to watch something or play on his switch. I totally leave him alone during this time and make sure his sister leaves him alone as well.
Making sure he knows exactly what’s happening each day and when.

Feel really sorry for him. I agree it sounds like he is masking and has tried so hard at school all day that he pops when he is home.

All my kids have been ruder at home than at school, because they are safer at home.

You don't seem to have much understanding of his disability. What is his diagnosis? I assume he has one as 6 is very young for an EHCP.