hum… 
it seems to me that “activity” is more like
https://your.domain/search?q=group%3Acamille.
Here, there is an intersection with category%3Acamille and filtering of what group members might have posted elsewhere.
And even more powerful (I discover something new every day 
), but you need the category_id
https://your.domain/g/camille/activity/posts?category_id=xxx
filters what is needed. I suppose automating this filter for the activity tab should be relatively “simple” 
, although… you still need to specify which category should be displayed for each group…