After extensive testing, I can say that the Invite Anyone plugin is preventing updates on groups.
After clicking 'Save Changes', the WP screen with message
Are you sure you want to do this?
I deactivated all plugins except BuddyPress, and GenesisConnect (I am using the Genesis framework and it is therefore necessary), and this plugin. The error appears when Invite Anyone is activated, and goes away when it is deactivated.
Using v1.0.15 of Invite Anyone, and BP 1.5.7
Is anyone else seeing this problem?