To not give her a lift?

Other Mum is total CF.
But this is the consequence of trying to use a polite excuse that a non CF would understand.
You're going to have to be more blunt.
Just say 'no'. You're not offering lifts. Tell the people at the venue when you arrive that you're not giving child X a lift at the end. Get your kid ready and leave.
CF should have been much more polite and asked you properly in the first place and taken a polite refusal

Don't give in Op, bugger what the other Mum's think. Message again and make it crystal clear, you weren't asked, the other Mum presumed and you don't do weekly favours for people who expect help but never offer in return.

Then ask because you dropped her child off last week can she drop yours off this week? If it takes a village it has to work both ways

I would just say something like, "no, sorry, but quite aside from the fact that I will be juggling the various things that I need to do of an evening, we tend to meet family after for dinner or whatever and that just wouldn't work. I agree that it is nice to help out, but if anyone had asked me I could have told them that I just can't do Wednesdays (insert day here). Perhaps for future terms we set up a lift share rota?"

Leave the group. Don't entertain these cfs

“As I said I can’t commit, but as you said it takes a village so thanks for volunteering to sort this out 👍🏻”

I also would be tempted to say I wasn't asked if I could provide a lift and I would have told you when I dropped your child off that this isn't happening again but only the lodger was home. I am not giving your child a lift. It doesn't work for me. Clearly there are others on here who believe in giving lifts so ask one of them to do it.

You have to stop this dead, I would also be notifying the place where the activity takes place that you will not be taking the child home so that they are aware and can follow their procedure for a child not picked up.

Just message "thanks for volunteering then, X. That is all sorted".

And just ignore the lot of them, this evening.

Stand firm, the others chiming in can help if they so desire.

OP said in her replies that she made the child sit in the back

Just message her privately and say it doesn’t work for you as her child misbehaved in your car.

I also have other commitments after the class, not just shopping, I can't commit to giving lifts after each class. Maybe we could take turns and you could bring the children home and drop mine off whilst I do my chores. Obviously, I would be willing to contribute towards travel costs, would that work for you?

If anyone else is willing to help us both out, please shout up.

That would be my response

I bet you anything she is out somewhere with others in the group. They have a pub night, or meal out and they need you to facilitate their plans. Something is going on here beyond the mass CFukery.

Mumsnet classic "sorry, that doesn't work for me" and nothing else.

You do not have to explain yourself and you definitely do not need to put yourself out for these people.

I'd also be tempted to say actually last time I did this there was noone home except the lodger, so if you're not available in future other arrangements need to be made.

I'd be fuming OP!

Respond ASAP clarifying that the arrangement is causing your child distress so you can't do it anymore. But since the rest of the group have mentioned the village you assume some of them will be able to step up and help

You're going to have to match her brass neck. I'd reply to the village comment with something like "What a nice idea, I'll be happy to share the load with the other 'villagers' and while we're at it, I could do with some help with XYZ. Any takers?"

As for her, say "I'm not committing to this, especially when you didn't even have the courtesy to ask me first. Find someone else."

I wouldn't even care if the rest of the group turned on me, not when the clear alternative is becoming her unpaid personal nanny.

Unfortunately offering an excuse is a mistake.
So the shopping excuse was a disaster.
We visit family afterwards if you feel you need to, is best.

Best to say "I'm afraid that doesn't suit me".

Ask where your village is then, its not like shes offering mutual support shes just taking without asking. I'd simply say it doesn't work with your child's needs and since the mum left her with you without so much as a please and the child hurt your DC and kicked your seat over and over you have no intention of being part of their village. Alternatively drop a link to this thread and hopefully they'll realise how much of a CF this mum is and how nasty their being to you.

I'd also ring who ever is running the activity and make it clear you will not be taking the child home.

The audacity of that reply!

Other posters have great replies but I feel like it needs to be pointed out to the group that none of this is asked, though it sounds like they'll side with her annoyingly!

"Since I wasn't asked about this which would have given me a chance to say in the first place, let me clarify that it doesn't work for me, so please ensure your child's journey home isn't my responsibility"

I would reply, if that is the case case I guess it is your turn.

How can she be in the front seat but also be kicking the back of your seat?

Anyway, message the mum to tell her it's not convenient - make up that you have another class to go to or something.

Perfect response 😁

As long as you have made it clear you are not taking this child do not reply further.

Speak to the session organiser and tell them ahead of the session that you will not be taking the child home so they are aware of the issue.

Do not allow yourself to be bullied.