Hi,
yes but this require some PHP and Joomla Devs knowledges.
Our delimiter field type allow you to use Joomla content plugins (docs about this field type at http://docs.easy-profile.com/index.php/article/html-delimiter-field-type).

You can use some free plugin like Sourcerer (https://www.regularlabs.com/extensions/sourcerer) to get usergroups and display with some code like this:

{source}

<?php

$owner_id=JRequest::getVar( 'id' , JFactory::getUser()->id ); 

$usergroups_ids=JUserHelper::getUserGroups( $owner_id );

$db=JFactory::getDbo();

$query=$db->getQuery(true);

$query->select( 'title' );

$query->from( '#__usergroups' );

$query->where( 'id IN ( ' . implode( ',' , $usergroups_ids ) . ' ) ' );

$db->setQuery($query);

$usergrous=$db->loadColumn();

echo implode( ', ' , $usergrous );

?>

{/source}

NOTE: this code is not tested and not supported, this is only for example purpose, you need to have PHP and Joomla devs knowledges.

Missing something

Also, it would be nice to have a search criteria or filter for users who only have a certain joomla group.

You can use SQL Custom Where to filter users in some list (for example to have users from some specific usergroups). See docs with examples at http://docs.easy-profile.com/article/sql-custom-where

Search features are not available for Joomla Usergroups.