You can do this by using the pre_get_posts
filter, which allows you to change they query before it is executed. In this way you can add Cat A to any request for categories. Like this:
add_filter ('pre_get_posts', 'wpse380282_add_cat_a');
function wpse380282_add_cat_a ($query) {
// do this only on the front end for the main query if it is a category archive
if ( !is_admin() && $query->is_main_query() && is_category() ) {
// get the ID for Cat A
$cat_id = get_cat_ID ('Cat A');
// get category of the query (returns an array of current categories)
$cats = $query->get( 'cat' );
// add Cat A to the array of current categories
$cats[] = $cat_id;
// set the category query to the expanded array
$query->set( 'cat', $cats );
}
}