You don't invite her. I tell AP we are going out with friends and to feel free to make her own plans including inviting a friend over to our house while we are gone. I don't give details of where we are going, make excuses, offer explanations, lie, etc. It is what it is. Your friends and extended family are not obligated to invite AP to everything. And you aren't obligated to invite AP to everything. You would never expect AP to invite you along to a party, dinner, movie, etc that AP is going to with her friends.
With that said, when we go just as family somewhere I invite AP most of the time. If we have a party at home, we include AP and tell her she can invite a friend or two. We always host a Thanksgiving with extended family and AP had always taken us up on the offer to invite a friend.