Yes, make a toolbar button which runs Go TABGROUPLIST=savecurrent,icons and it will show a list of all tab groups, and save the current group (if any) any time you change to a new group using the list it generates.
Note that it won't save the group when you close the window or exit the program (without loading another group first). That would require some extra scripting, but can be done as well if needed.
EDIT Nov 2023: Added ,icons to the suggested command, so each group has an icon.