the thing to ask yourself is how you end up in this situation - not in a judgmental way but more analytically. My first question in the situation written out in your OP, I would point out you said you were "hoping to befriend her." Implying that you may not know this person very well in the first place. How well did you know her before inviting her into your group? Could it be perhaps you're rushing into joining groups, meeting people, befriending them and such and not allowing things to happen naturally ? I'll tell you a secret, no one "effortlessly" makes friends. Not ones that count anyway, and sure there are the socialites that are very much outgoing and it may seem to be that way but if you were to look closer, they are social and outgoing but many if not most of their connections are very thin. Almost all people really do only need and have a couple of really close friends and that's really all you need.

Stick to the groups, but don't get too caught up in making connections instantly or even quickly. It takes time to get to know people. Let it happen as you become a regular and everyone becomes acquainted with you and vice versa. Don't quit because of a few obnoxious turds (pardon the term but I think it's fitting). Because there are always a couple or even a few of them anytime there is a group of any number. Let that go, enjoy the groups with things you enjoy and let friendships happen as they come.

It can't be forced, there is no secret formula and the only thing you "may" be doing wrong is quitting and giving up too soon

Hang in there.