This is an awesome problem to be looking into. Thanks for raising it, @Judith! Groups are so powerful and have so much promise! 
I noticed this issue recently as well, when looking at maintaining group membership from wordpress. It would be pretty neat to be able to automagically add/remove people from groups for country and region of the world they say they are based in, languages they speak, themes they are interested in, etc, when they sign up and update their profiles in wordpress.
@justin’s suggestion of storing both group name and group ID is a good one and I’ll pass it on to the folks working on this integration now.
One feature I’d also love to see added to the UI is the ability for admins or group owners to bulk remove all members from a group, and re-add them. This would allow them to reset their group memberships manually from time to time based on exports from wordpress, even if they don’t have an automated systme in place to add/remove people from groups as they update their profiles in wordpress.