Not unless you include that asset in the activity tab asset param. The best approach would be to utilize the new Follow API and add a Follow button to GJ so they can follow the group and optionally have its activity display on their profile. Added a feature ticket to add a new system action to CB Auto Actions for outputting a follow button in groups.
forge.joomlapolis.com/issues/6858
The below for example should work to add a follow button to groups.
Global
Triggers: gj_onBeforeDisplayGroup
Type: CB Activity
User: User
Access: Everybody
Action
Mode: Stream
Stream: Follow
Asset: groupjive.group.[var2_id]
Output
Display: Silent
Layout:
Code:
$variables['var6'][] = $content;
Method: PHP
Parameters
Reference Variables: Variable 6
That'll allow anyone to follow the group. You can customize it further with various conditions as well.