Activity level permission

Is it possible to show an activity A1 to user group A and another activity A2 to user group B ?

Activity does not have ACL. You'd need to have 2 different streams with 2 different Assets specific then adjust the access to that stream using the ACL of the module if output as a module, he menu item if output as a menu item, or the field if output as a field.

Sorry, how do I create two streams ?

You've 4 options for doing that. You can create a Joomla menu item with Community Builder > Plugin menu type then select CB Activity as Plugin and set Action to Activity; now under the Activity tab add your custom Asset. You can basically do the same as a Joomla module using the CB Activity module. Finally you can also use the Activity fieldtype to have the custom stream be a field displayed on profile. Finally you've CB Activity Bot that lets you substitute a stream anywhere that Joomla content prepare is ran (e.g. in a Joomla article), but that's a bit more complicated to use compared to the other 3 options.

How to use the 'Asset' to show activities arising from a particular user group ?

e.g. I want to list only activities from a particular user group.

Activity does not have ACL. It can not display posts only from those in a specific usergroup. The only way to do that is to have a stream that only allows that usergroup to post on it using the create access parameter or limiting access to the entire stream it self. Example as follows.

Asset: custom_stream
Create Access: Author

This should output a unique stream that only displays activity with an asset of custom_stream and only allows authors to post on it. You can use whatever alphanumeric string you like for asset.