How can one Rewrite results of a view using a Group By
You can use both Rewrite and Grouping at the same time. You can Rewrite the field and use that as Group by. 

If you need the same Field in the results as well as in the Group by headings, then simply add the Field twice. You can style/Rewrite them separately too. 

If you want to style the grouping in a special way, or need more than one field in the Group, then you can use the Custom text field for advanced styling of the Group. You can see an example in this article: Control Group by styling with Custom text field