How can you embed the Wordpress Groups widget inside a page/publish outdoors the sidebar?

I believe, you look for this.

http://wordpress.stackexchange.com/questions/36511/are-widgets-meant-to-be-used-outside-of-sidebars

To include wordpress publish groups in to the template file you should utilize

<?php wp_list_categories($args); ?>

Default arguments are :

<?php  $args = array(
      'show_option_all'    => ,
      'orderby'            => 'name',
      'order'              => 'ASC',
      'show_last_update'   => 0,
      'style'              => 'list',
      'show_count'         => 0,
      'hide_empty'         => 1,
      'use_desc_for_title' => 1,
      'child_of'           => 0,
      'feed'               => ,
      'feed_type'          => ,
      'feed_image'         => ,
      'exclude'            => ,
      'exclude_tree'       => ,
      'include'            => ,
      'hierarchical'       => true,
      'title_li'           => __( 'Categories' ),
      'show_option_none'   => __('No categories'),
      'number'             => NULL,
      'echo'               => 1,
      'depth'              => 0,
      'current_category'   => 0,
      'pad_counts'         => 0,
      'taxonomy'           => 'category',
      'walker'             => 'Walker_Category' );
 ?>

or simply employ it without parameters, for the reason that situation defaults works.

 <?php wp_list_categories(); ?>

read much more about this function here