I am not that good of a coder. The problem is that even if they are in only one group, any code that you write has to allow for the possibility of multiple groups.
Basically, you would have to find the code for the grader report and change that column to pull the group info (however, it is going to have to pull it from a different table which adds another layer of complexity.)
You can already filter by groups in the grader report (dropdown list at top of page)- what I do is put that information in a field that the grader report can read like Institution or Department which you already have there. It really is simpler if you keep it to pulling information from the user, not from the course.