No solution yet, although a clever system admin might be able to create a workaround script or tool using the REST API - if they could find a way to make it executable without prompting for an admin password.
If you'd like to be able to have someone manage membership in a group without promoting them to an admin then vote up these ideas. Some solution to this issue is long overdue IMHO. Admins have enough to do without managing groups, and the data owner is usually in a better position to decide which users should belong to their group in any case.
Thank you, Alex. Finding similar issues was actually quite difficult due to the commonality of the keywords involved--despite diligent searching I could not find any of those linked posts.