1. Create a web-site (community)
2. Create a group
3. Add user to group
4. Add group to site
5. Do something within that site, that triggers the creation of an activity.
6. Place Activity Portlet on users private Page
7. Click to my-sites tab of the activity portlet
You will not see the activity created in step 5.
It works well, when the user is a direct member of the site.