Can someone resolve second problem:

I wanna list all groups on primary page of this site, and every category can have only 4 last posts with title and favorite image.

How to get this done, assist me to, please :)

<?php  $cats = get_categories('orderby=count&order=DESC'); 

foreach ($cats as $cat) :

  $args = array(
  'posts_per_page' => 1, // max number of post per category
  'cat' => $cat->term_id
    if (have_posts()) : while (have_posts()) : the_post(); ?>
    <!-- show the ususal stuff in the loop -->
    <?php endwhile; endif; wp_reset_query(); ?>

<?php endforeach; ?>