This is unfortunately still broken and I imagine is due to a single directory having the wrong permissions (although perhaps it needs to be changed for every groups’ directories). I really hope a fix is coming soon.
Oh, I’m sorry, I should have clarified: this happened on buddypress.org when trying to upload an avatar. So I don’t have access to the server. I imagine something is wrong on this server, assuming that I should be able to upload a group avatar…