The group activity is a great idea since you're with that person, but technically with others as well, so you're out together but technically out with friends as well. Not quite sure why I didn't work that out earlier!
In practice though there are a few "social challenges" to make that happen though. Firstly we are not in the same group of friends at high school, she has a mostly girls group and we have a mostly guys group, we only have one mutual friend who usually doesn't sit with either of us and has her own group of friends too.
So from either my end or hers, asking our group to do a joint activity with the other group is going to be treated as weird or suspicious. I know for my friends, especially if they're not interested in anyone from that group, the first thing they'll think is that I'm doing it because I like someone from that group, unless it is an event where we all happen to be there randomly eg. school event/excursion. So if I was my friends I would be thinking: so you like this girl and you want to drag all of us out to make a move, what's in it for me? It doesn't bother me that they would know I'm interested, but it feels a bit self serving that's all. And I can't talk for her, but I suspect a group of girls being asked to a social activity with a group guys they don't usually talk to will be treated awkwardly too. Unless of course we invited just our friends without telling them who else was also going, a bit sneaky though, and not sure it would work after the first time.
It almost seems like it would be easier if we were official because then our intentions are clear to both groups.