How to add button to top of theme customizer?

If you take a look at the source code of this part …

do_action( 'customize_controls_print_styles' );
do_action( 'customize_controls_print_scripts' );
?>
</head>
<body class="<?php echo esc_attr( $body_class ); ?>">
<div class="wp-full-overlay expanded">
    <form id="customize-controls" class="wrap wp-full-overlay-sidebar">

        <div id="customize-header-actions" class="wp-full-overlay-header">
            <?php
                $save_text = $wp_customize->is_theme_active() ? __( 'Save &amp; Publish' ) : __( 'Save &amp; Activate' );
                submit_button( $save_text, 'primary save', 'save', false );
            ?>
            <span class="spinner"></span>
            <a class="back button" href="https://wordpress.stackexchange.com/questions/96947/<?php echo esc_url( $return ? $return : admin_url("themes.php' ) ); ?>">
                <?php _e( 'Cancel' ); ?>
            </a>
        </div>

… you can see there is no action for this place.

But you can hook into customize_controls_print_scripts and register a new JavaScript that inserts a button.

add_action( 'customize_controls_print_scripts', 'add_customizer_button' );

function add_customizer_button()
{
    wp_register_script( 'my_customizer_script', plugins_url('/customizer_button.js', __FILE__) );
    wp_enqueue_script( 'my_customizer_script' );
}