I think there is a misunderstanding (if I am not mistaken): A group is linked to users (a user attribute: they are either part of a group or not), whereas a category applies to discussion topics (which category each topic is located in). I don’t think you can “put groups into a category.” Unless you mean that you created categories whose access is restricted to specific groups?
You are seeing the activity of the group members. A group is simply a selection of members. What do you mean by an activity “related to the group”? Only what happens in a category whose access is limited to the group?
(Provided I have understood correctly)