Ok, so I Social Shared a group (keep in mind the group is Invite Only) so I understand this part.
However, my opinion is if I share a group (ANY) group it should pull the Group Logo and the Group Description for the feed.

Otherwise what is the point?

Even on a Public Group it shares the most recent post in the stream but has no link back to the group, group name, nothing. It is also pulling the main site Description into the FB share.

See Example: http://screencast.com/t/wbdkfzw0ZdeK

Thanks,
James