@Snow34 Don't use the phrase help, he doesn't help put the kids to bed, that implies it is your job. It is called parenting a child. He should be parenting.
How old are the children? What chores could they be doing in an age appropriate way? If they can toddle, laundry into a basket. If they are at school they can set a table, load plates into the dishwasher, wipe the table, possibly strip a bed, make up a bed. Lots of children at boarding school do this every week. Not everything falls to the Mum. This isn't the 1950s.
When children come home from school they put their things away, just like they are expected to do in school, coat where it belongs, shoes away, empty their lunch bags if they have them etc. They tidy up after themselves, this starts when they are toddlers. This is a team, the team works together to keep the house tidy. We used to sing a theme tune to a tv show about team work. We'd also chant Team (insert surname or a team name)
Equal free time, why is his free time more important than your free time? I say this as a long term SAHM with a completely plugged in, hands on husband. What sort of role model is your Dh setting for the children? That Dad's get to sit on their arse or those that work full time? Just trying to see his thinking.