Make custom post type column sortable

I believe the missing piece to your puzzle is one additional filter to perform the sort and return the sorted data. Try something like the following:

function sort_column_by_auction_1357( $vars ){
if ( isset( $vars["orderby"] ) && "auction" == $vars["orderby"] ) {
    $vars = array_merge( $vars, array(
        "orderby" => "auction"
    ) );
}
return $vars;
}
add_filter( "request", "sort_column_by_auction_1357" );

This worked for me. I added a sort by Last Modified to all of my custom post type views.

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