[Thomas M]

The following code works for me:

function automatic_group_membership( $user_id ) {
   if( ! $user_id ) { 
	   return false;
   }
 
   groups_invite_user( array( 'user_id' => $user_id, 'group_id' => $group_id ) );
}
add_action( 'bp_core_activated_user', 'automatic_group_membership' );
add_action( 'user_register', 'automatic_group_membership' );

You need to replace $group_id with the ID of the group you want to assign users to.
I have not tested the code when inviting members as a non-admin. That might be a limitation. There is an optional is_confirmed argument that can be submitted with groups_invite_users. Maybe, setting it to true helps in such cases.