Humm there might be something I miss behind the model (which make sense since avatar is not in table...) to get a group category avatar...
$catModel = FD::model('GroupCategories');
$b = $catModel->getCategories(array('id' => $category->esgroup_id, 'state' => SOCIAL_STATE_PUBLISHED, 'ordering' => 'ordering'));
$a.='<img src="'. $b->getAvatar(SOCIAL_AVATAR_SQUARE).'" class="avatar" />';
$b retrieve the good category... however getAvatar method fails... :/ which makes sense, how I link $b and getAvatar ?