Mrs. Kingsley's son, Freddie, goes to Abbey Mount's brother school.
Speaking as someone who never went to a single-sex school or any kind of private school (day or boarding) or any school in Europe, I'm not certain as to how this would work in England, but I imagine that most all-girls schools (whether in or outside of the United States) has a brother school of some kind.Mrs. Kingsley is a former student at Abbey Mount before she eventually become the school's headmistress.
Not only that, but I also think that Mrs. Kingsley was a student at Abbey Mount at around the same time as Poppy's mother—while they may have not been in the same grade/graduating class, they were at least schoolmates.