Get list of nodes by field-collection value
I think you have to possibilities to solve this:


Looks like you already got a tag for joining the field_group entities here: 

$query->addTag('joinentity');


You could then implement hook_query_joinentity_alter() to implement the join, see also EntityFieldQuery INNER JOIN


You use db_select() or db_query() and add your joins there rather than an EFQ. I think this is preferable as implementing tags and hooks for those gets a bit messy IMO.