Getting group posts in og's home page view
I recommend creating a whole new view, which would list the three most recent posts for the group passed in the argument.

To make it work together with your existing og view, you'll want to attach the new view to your group content type using one of Viewfield, Views Field View and View reference modules.

For example, using Viewfield; to the group node type add a hidden View field called Recents. Set it to "Always use default value". Set the default value to the recent posts view. The argument is "%nid". Exclude its display from node teaser, full view etc.

In the group view, then, along with group name and mission, also add Attrview as a field. Good luck!