Please go to class-fep-group-message.php line 320 and change return $user_groups; to return apply_filters( 'fep_get_user_groups', $user_groups, $user_id ); (this hook will be added to next version, so you will not be needed this change for next version)

Then add following code in your theme’s (child theme’s if any) functions.php

add_filter( 'fep_get_user_groups', function( $user_groups, $user_id ){
    $groups = fep_get_option('gm_groups', array());
    if( $groups && is_array( $groups) ){
        foreach ( $groups as $group ) {
            $user_groups[ $group['slug'] ] = $group['name'];
        }
    }
    return $user_groups;
}, 10, 2);

This will allow you to send message to any group without joining that group.

did you replace code exactly what i gave you? Please recheck.
If still not working please give me screenshot of what you have changed.
Also let me know your PHP version.

Well done! Thank you for this fix. I have made these additions and can confirm this works.