Make Wordpress subcategories use Category Template

Question

I've got a category template: category-projects.php

This category has subcategories, but they're refering to the template category.php for instructions instead of the parent category. How do I make subcategories refer to parent category templates in the cascading order of template references?

*Note, I'm talking about category level urls, not posts.

Answer 1

Richard's answer does work but it will heavily interfere with other plugins.

I found a better alternative using add_filter & template_include as the example below

add_filter( 'template_include', 'my_callback' );

function my_callback( $original_template ) {
    if ( some_condition() ) {
        return SOME_PATH . '/some-custom-file.php';
    } else {
        return $original_template;
    }
}

Credit to https://markjaquith.wordpress.com/2014/02/19/template_redirect-is-not-for-loading-templates/