To not give her a lift?

Sorry. I am not able to do that.

Simple, direct. Leave it at that.

How was she kicking the back of your seat if she'd manoeuvred her way into the front?

This.
And I cannot understand why the organisers are allowing random parents (no offence intended OP) to take just any child!!! The whole set-up sounds amateur and unregulated !!!! They are children FFS.
Plus, had it been me giving the child a lift home to a house with no parents there - I might have taken the child to the police station for safeguarding reasons. Entitled mother would think twice next time.

I wouldn't have given a reason.

Something like "Hi folks - I am unable to provide lifts either to or from X activity. Please don't ask as the answer will be no, I can't."

By saying that you do your shopping someone may come back and say "Sure Adam doesn't mind a bit of shopping of a Tuesday evening. Thanks for taking him" or some such nonsense.

Don't give them more information than you have to, and I think you've given them more information than you had to or needed to.

There’s no silver bullet with this one it’s just going to have to be a direct no

This! And stand firm.

If the mother is not there to pick the child up, the child will have to stay with the leaders until they do.

it’s not your problem to resolve.

Does NOT cover passengers.

Please never take the word of an 10/11 year old that you have permission from their parent to do anything.
What if the OP had not been there that evening?
What if the parent had turned up five mins later and her child was gone?

Anyone who runs a club would know this is a safeguarding risk and any decent activity leader would not allow this to happen without speaking to parent.

Obviously now it appears that OP does have permission, so it's up to her what she does next.

WTF?! You don't have to justify or explain yourself to any of these people. Just say you aren't doing, personally l would mention the rude way you were informed you were doing it and bratty behaviour of the child. Other mums can give them a lift home if they are that bothered

Say your insurance does cover passengers.

Message the mother directly. Just say you won't be giving any lifts in the future - don't make any excuses as there is not need to explain your reasoning. You have absolutely no responsibility for her child. Phrase it politely but firmly.

I wouldn’t have brought her home the first time. It’s incredibly entitled and irresponsible of her. Just ignore and leave her there and let the organisers deal with mum.

This is really good

The amount of people who think this is a gotcha is weird.

By offering excuses and solutions, you’re embroiling yourself in a situation that has nothing to do with you.
Don’t excuse or elaborate.
Just a simple no, and walk away.
They will kick off. Let them.
Read or listen to The Let them Theory by Mel Robbins, life changing.

Don't add information or excuses it just gives her more room to pressure you into saying yes.

Reply. That doesn't work for me. I cannot give your child a lift home.

Where does OP say the child was in the front seat?

Feck that!

Go to the organisers of the after school activity (whatever it is) and tell them that you were left in a very difficult position after the activity on X date where the child was under the impression that you would be bringing them home. You have no idea where the child would have gotten that impression as you have never spoken to their parents and no one even asked you whether it might be possible to do this, so as a safe guarding issue, you are bringing it to their attention.

Tell them that you are not able to bring this other child either to the activity or home from it as you have your own child to attend to and it will not be possible even if this other parent asks you. It's just isn't possible.

That should cover your own child's safe space after the activity.
It should also cover the safeguarding that is required here.

If it happens again, I'd bring the child back into the activity hall and hand them over to the organisers. Not your child, not your job to get them home.

Did the child sit in the front as you say or behind you where they were able to kick your seat ?

Just tell her that her child was so disobedient and badly behaved in the car last week that you will not be taking her food shopping with you so she will need to make other arrangements.

This seems absolutely crazy that a parent would expect you to give their child a lift if they'd never even spoken to you (thinking from a safety aspect, how do they know they can trust you).

Tell venue you have no knowledge of the arrangement and are not able to take the child home. They're not going to be happy with the parent, will have to contact them and they'll have to arrange transport themselves.

She shoved OP's child out of the way to get in the front seat but OP didn't let her. She got in the back and obviously kicked OP's seat the whole way out of frustration due to not getting her own way.

I was afraid she would say that, I bet if you said you were going to a cafe for tea, or to GPs for the evening she would say she'll go there too. They seem not to be home anyway so don't really care where she is. Say you can't cope with another child while out shopping or take that responsibility, dangerous car parks, child wandering off in the store.

How was she kicking the back of the seat from the front of the car?
Really weird if you don’t know the parents. How old are they?

I’d go to the mum directly and tell her you cannot do lifts home as you actually don’t go home after the activity so will not be passing her home - and had she had the courtesy to ask you this at the outset you’d have told her this.

If she pushes back, state that you are not an uber service and don’t gift lifts to random strangers’ children.

I’d also, before then, drop a note to the activity organiser to explain that there seems to have been an assumption made on the part of this parent and that you will not be taking this child home after the session, so can they ensure that the mother is aware of her responsibility to collect her child. For safeguarding purposes, they will not want a child left in their care because you have not taken them home, so will hopefully step in if needed.