While trying this request
$result = civicrm_api3('Contact', 'get', array(
'return' => "id",
'group' => "Administrators", ));
we've got
{
"error_code": "no such field",
"sql": "SELECT contact_a.id as contact_id, contact_a.is_deleted as `contact_is_deleted`, `civicrm_group_contact-Administrators`.id as group_contact_id, `civicrm_group_contact-Administrators`.status as status FROM civicrm_contact contact_a LEFT JOIN civicrm_group_contact `civicrm_group_contact-Administrators` ON (contact_a.id = `civicrm_group_contact-Administrators`.contact_id AND `civicrm_group_contact-Administrators`.status IN (\"Added\")) LEFT JOIN civicrm_group_contact_cache `civicrm_group_contact_cache_Administrators` ON contact_a.id = `civicrm_group_contact_cache_Administrators`.contact_id WHERE ( contact_a.is_deleted = '0' AND ( ( `civicrm_group_contact-Administrators`.group_id IN ( Administrators ) ) OR ( `civicrm_group_contact_cache_Administrators`.group_id IN (\"Administrators\") ) ) ) AND ( 1 ) AND ( 1 ) GROUP BY contact_a.id LIMIT 0, 25 [nativecode=1054 ** Unknown column 'Administrators' in 'where clause']",
whatever operator ( "=" , "Like", "In" ) is being used.
All we wanted is Contact IDs, which should be passed to another API call. Of course, we can get data by Search in Group in browser and export all values, when read csv file and extract IDs, but it would be much prettier with pure API calls.