It should work in the same way that saving any group currently works - that is, across all apps. If a group tab has been named or renamed, the save should capture the assigned tab name, so that when accessing the saved group the tab names are there. At the moment, the renaming of tabs only holds for the current session, when accessing the saved group the tab names are lost, which for me defeats the whole purpose of saving the group.
Personally if find this most annoying when using Groupy to save browser windows, where I might have several tabs named for open browser windows e.g. Work, Personal, Research... etc. Save the group, next time, names are lost, so the 'save' isn't really a 'save'.
Great app, except for this seemingly obvious omission. Thanks for giving this some attention.