Have you tried this?
'meta_query' => array(
'relation' => 'AND',
array(
'relation' => 'OR',
array(
'key' => 'start_date', /// or 'end_date' for cpt 'group_of_events??
'compare' => 'BETWEEN',
'type' => 'DATE',
'value' => array($filter_start_date, $filter_end_date),
),
array(
'key' => 'end_date', /// or 'end_date' for cpt 'group_of_events??
'compare' => 'BETWEEN',
'type' => 'DATE',
'value' => array($filter_start_date, $filter_end_date),
),
),
array(
'key' => 'type',
'compare' => 'IN',
'value' => $filter_type,
),
),
then you can change the start_date of the events to another key.