Wordpress - Extending the search context in the admin list post screen

I solved filtering the query by adding the join on the postmeta table and changing the where clause. tips on filtering the WHERE clause (often require regular expression search&replace) are here on codex:

add_filter( 'posts_join', 'segnalazioni_search_join' );
function segnalazioni_search_join ( $join ) {
    global $pagenow, $wpdb;

    // I want the filter only when performing a search on edit page of Custom Post Type named "segnalazioni".
    if ( is_admin() && 'edit.php' === $pagenow && 'segnalazioni' === $_GET['post_type'] && ! empty( $_GET['s'] ) ) {    
        $join .= 'LEFT JOIN ' . $wpdb->postmeta . ' ON ' . $wpdb->posts . '.ID = ' . $wpdb->postmeta . '.post_id ';
    }
    return $join;
}

add_filter( 'posts_where', 'segnalazioni_search_where' );
function segnalazioni_search_where( $where ) {
    global $pagenow, $wpdb;

    // I want the filter only when performing a search on edit page of Custom Post Type named "segnalazioni".
    if ( is_admin() && 'edit.php' === $pagenow && 'segnalazioni' === $_GET['post_type'] && ! empty( $_GET['s'] ) ) {
        $where = preg_replace(
            "/\(\s*" . $wpdb->posts . ".post_title\s+LIKE\s*(\'[^\']+\')\s*\)/",
            "(" . $wpdb->posts . ".post_title LIKE $1) OR (" . $wpdb->postmeta . ".meta_value LIKE $1)", $where );
    }
    return $where;
}

Stefano answer is great but it lacks a distinct clause :

function segnalazioni_search_distinct( $where ){
    global $pagenow, $wpdb;

    if ( is_admin() && $pagenow=='edit.php' && $_GET['post_type']=='segnalazioni' && $_GET['s'] != '') {
    return "DISTINCT";

    }
    return $where;
}
add_filter( 'posts_distinct', 'segnalazioni_search_distinct' );

Add the code above update it and it will work with no duplicates.

This will work,

function custom_search_query( $query ) {
    $custom_fields = array(
        // put all the meta fields you want to search for here
        "rg_first_name",
        "rg_1job_designation"
    );
    $searchterm = $query->query_vars['s'];

    // we have to remove the "s" parameter from the query, because it will prevent the posts from being found
    $query->query_vars['s'] = "";

    if ($searchterm != "") {
        $meta_query = array('relation' => 'OR');
        foreach($custom_fields as $cf) {
            array_push($meta_query, array(
                'key' => $cf,
                'value' => $searchterm,
                'compare' => 'LIKE'
            ));
        }
        $query->set("meta_query", $meta_query);
    };
}
add_filter( "pre_get_posts", "custom_search_query");