Include customer details in woocommerce 3rd part plugin email

Either you work with a hook

function include_customer_details( $order, $sent_to_admin, $plain_text, $email ) {
    // Only admin need to know.
    if ( !$sent_to_admin ) {
        return;
    }

    // test first, then delete the line below and complete the if statement
    echo '<pre>', print_r($email->id, 1), '</pre>';

    /*
    if ( $email->id == 'paste result from test mail' ) {    
        echo 'my extra text';
    }
    */
}
add_action('woocommerce_email_customer_details', 'include_customer_details', 10, 4 );

Or adjust the template file, because you have access to $order you can get easily order info

More information:

https://businessbloomer.com/woocommerce-easily-get-order-info-total-items-etc-from-order-object/

<?php
/**
 * Admin auction finish  email
 *
 */

if ( ! defined( 'ABSPATH' ) ) exit; // Exit if accessed directly 
$product_data = wc_get_product(  $product_id );

?>

<?php do_action('woocommerce_email_header', $email_heading, $email); ?>

<p><?php printf( __( "The auction for <a href="https://wordpress.stackexchange.com/questions/359270/%s">%s</a> finished. Winning bid is %s. ", 'wc_simple_auctions' ),get_permalink($product_id), $product_data->get_title(), wc_price($product_data->get_curent_bid())); ?></p>

<p><?php echo '<pre>', print_r($product_data, 1), '</pre>'; //etc... ?></p>
<p><?php do_action( 'woocommerce_email_customer_details', $order, $sent_to_admin, $plain_text, $email ); ?></p>

<?php do_action('woocommerce_email_footer', $email); ?>