There are 2 attributes on the Support Group window: Name & Title.
You can change the name without any issues, b/c the unique identifier (title) will remain unchanged.
As long as your queries are built on the Support Group.Title field, you should be ok.
You may want to go into query and report designer and change it to "View By" module / Business object.
Check all your queries, filters, etc that are built on the System\Support Group module/Business object, and make sure that any criteria are built on the correct attribute: Name or Title.