I'm trying to imagine how any of this conversation has gone the way the OP describes it. It just makes no sense to me.
One of the parents, who I have never spoken to, has asked other parents where I live, and on discovering that I drive past her house, has decided that I will be taking her child to and from the activity from now on. She has not asked me this!
So last week it seems this child just announces to the OP 'my mum says I am getting a lift home with you tonight. My address is XXXX.' with no prior discussion.
If that happened to me, I'd be asking the child to get their mother on the phone to make sure I wasn't inadvertently kidnapping someone's child, for a start. If there was some sort of issue, I'd help out for that one occasion rather than leave the child stranded with no immediate lift in place, given that home was 30 mins away. But the OP just does it, and leaves the child at the house with an adult who wasn't even a relative.
the mother had messaged the other parents on the group chat and told them I was taking her child home.
That's it?
On a WA message group that the OP is part of, this woman has said something along the lines of 'Dreaming is going to be taking Amelia to club and bringing her home again this evening and from now on.' Without ever actually asking Dreaming herself if this is okay and without having had any contact with her at all since she brought the child home last week. Seriously, who does that?
Why would they bother to announce this on a chat group to a load of other people who don't even need to know, while talking about the OP in the third person as though it's already been discussed/agreed between the two of them, when they've never actually spoken AT ALL before?
Surely it would be just as easy to say 'Dreaming , I hope it's okay for Amelia to jump in with you again tonight? That would really help me out a lot. Thanks.'
And then:
'I have put a message on the group chat saying I am unable to give lifts home as I do my grocery shopping after the activity.'
Which is fair enough. But then the OP has said the other mums on the chat seem to think she's being unreasonable and two have actually gone as far as to take her up on it, with the comment 'It takes a village to raise children.'
WTAF? Is this a bad episode of Desperate Housewives?
Is there some mitigating factor here, like the child's mother is recovering from emergency surgery because she was hit by a juggernaut last week or something? Because otherwise I am really struggling to imagine this conversation playing out the way the OP describes it.
Sometimes people are just bad at describing things and either add stuff for effect but make the whole thing sound far less less believable than the truth, or they leave out important information that gives better context. As it is, I just can't get my head round this happening at all.