I create template customsearch.php, adding this for page example\search.html (ID page: 18)

    <?php
    /*
    Template Name: CustomSearch
    */
    
    get_header(); ?>
    
    <div id="primary">
            <div id="content">
        
    <script async src=".js?cx=000000000000000000000:uiyuiyiyiuyui"></script>
    <div class="gcse-search"></div>
</div>
</div>
    
    <?php get_sidebar(); ?>
    <?php get_footer(); ?>

How to this page adding for example?q=xxxxxx

Result:

example/?s=xxxxxx - default search wordpress

example/?q=xxxxxx - google custom search

Sorry for my English

P.S.

I want to keep a default search