I had a similar problem with another mother back a couple of years ago. It didn't even have to involve her own kid. Both of our daughters went to the school's aftercare program. Other Mom emailed me asking if everything was okay, because she had seen that my DD and another girl (not her daughter) were upset with each other when she went to pick up her daughter from the afterschool program. She had asked the program director what was going on and he gave a polite non-answer, since it wasn't her kids. So then she asked her daughter about it, and gave me this whole kind of rant about how it made her daughter uncomfortable when her friends fought and the girls all needed to get along because they were in girl scouts together and we should sit them down and tell them how this was a big deal. I responded that I didn't know what had happened, but that when I picked up my DD that day (about 15 minutes after other mom had been there) she and the 3rd girl where playing a game together and the program director hadn't said anything. I also pointed out that we had never had a problem with the girls at Girl Scouts, so I didn't think an intervention was necessary. That night I asked DD about it, and it turned out the girls had got into a tiff over something, but they apologized to each other. I asked about the Other Mom's daughter's involvement and DD said she wasn't even there, she was playing with another group of kids on the playground.
After that it just kept coming....I'd get reports from other mom about DD doing something like not playing with her daughter. When I'd ask DD, it would turn out the whole story was that they invited girl to play some game with them and she didn't want to and instead tried to get them to all do what she wanted, then got mad when they didn't.
My first response was to tell DD to distance her self from this girl. Not to make it obvious, but just find someone else to play with. That helped for a few weeks, but then the Other Mom emailed me and basically accused DD of bullying and had some story about DD and the other girls forming a club and they had a list of members that all the girls in the after school program were on the list except her daughter. I asked my DD about it and asked to see the list. Their "club" was 3 or 4 girls talking about American girl dolls and their "member list" was a list of AG dolls that they all wanted. This girl wasn't into the dolls, so she opted to play basketball instead.
That was the final straw for me. I told the mom that I thought it best that the girls not play together and just avoid each other. Other Mom agreed to that.
What I wish I would have done is said something to the after school director and to the school counselor, because that girl did complain to the counselor and DD had to go talk to her about it. The counselor was cool about it, DD didn't feel punished or anything, but I was not happy. Fortunately it was the last week of school, so I just let it go.
In talking with the other mom, she admitted that she was picked on a lot when she was in school. I think her daughter learned pretty quickly that if she said people were picking on her, her Mom would be very quick to defend her. Other Mom did finally catch on that her daughter wasn't being completely truthful, so things have improved.