Sarah and Andrew are divorced, but they have lived together for many years. They are domestic partners. The late Duke of Edinburgh disliked Fergie , but the late Queen was fond of her, so Fergie would be invited to Balmoral, Windsor, etc when Philip wasn't there. After Philip died, Fergie was invited to more family things and that has continued. I doubt anyone in the family is fond of her, but they do like and support her daughters and their families, so she gets invited as Andrew's partner, not on her own merits. And I think Anne and Edward want Andrew to be happy.