How to set up Author archives with sub category URL

As I mentioned in my comment, you can use add_rewrite_endpoint to accomplish this.

First, add the endpoint:

function wpa_author_endpoints(){
    add_rewrite_endpoint( 'liked-posts', EP_AUTHORS );
}
add_action( 'init', 'wpa_author_endpoints' );

After flushing rewrite rules (visit your Settings > Permalinks page in admin), author URLs can now be appended with /liked-posts/.

Next, add a filter to author_template to load a different template for these requests. This checks if the request has set the liked-posts query var, and loads the template liked-posts.php if it exists:

function wpa_author_template( $template="" ){
    global $wp_query;
    if( array_key_exists( 'liked-posts', $wp_query->query_vars ) )
        $template = locate_template( array( 'liked-posts.php', $template ), false );
    return $template;
}
add_filter( 'author_template', 'wpa_author_template' );

Within that template, you can use get_queried_object to fetch the author data for the queried author, which you can use in additional queries to load your author data.

EDIT – pagination doesn’t work with an endpoint, because anything after the endpoint gets put into the endpoint query var. so to get the page number, just extract it from the query var:

if( strpos( $wp_query->query_vars['liked-posts'] ,'page') !== false ) {
    echo substr( $wp_query->query_vars['liked-posts'], 5 );
}