I ended up deciding to let her go ahead with it. This is what I told her:

As long as you put a note at the beginning saying that it's based on my story but it's YOUR continuation and not necessarily how I would continue the story, I have no objections. That means I don't want you to repost my story as your first chapter -- just point readers to it, please. I don't need to beta read it (in all honesty, I'm a terrible beta and I'd probably drive you nuts tweaking things) but I'd appreciate a notice when you post your story.

I think that gives her free reign to go where she wants while making it clear that I'm not really associated with whatever she writes. (In case it ends up being horrifically awful, y'know.)