BuddyPress.org

It’s just not a core issue with BuddyPress and dives into the customization of BP. Unless someone has specifically needed to do this in the past no one will be likely to help you.

I would either put out a call to hire if you can not get it to work or possible post at woo or try something like http://stackoverflow.com

It really shouldn’t be that hard. Create a cpt loop on the front page of the group and then where you call the post type return the group name that matches the cpt

$cpt = some code to get the group name
$args = array( 'post_type' => $cpt, 'posts_per_page' => 10 );
$loop = new WP_Query( $args );
while ( $loop->have_posts() ) : $loop->the_post();
	the_title();
	echo '<div class="entry-content">';
	the_content();
	echo '</div>';
endwhile;