Helps you to customize the login & registration forms in Tutor LMS WordPress plugin.

override-tutor-login-registration-form.php

<?php

/**
 * Fires Before Tutor Global Login Form
 * @Hook: tutor_load_template_before
 * @Template: global.login
 */

add_action('tutor_load_template_before', 'tutor_global_login_before', 10, 2);
function tutor_global_login_before($template, $variable){
    if ($template === 'global.login'){
        echo '<button>Login with Facebook</button>';
    }
}


/**
 * Fires After Tutor Global Login Form
 * @Hook: tutor_load_template_after
 * @Template: global.login
 */

add_action('tutor_load_template_after', 'tutor_global_login_after', 10, 2);
function tutor_global_login_after($template, $variable){
    if ($template === 'global.login'){
        echo '<button>Login with Facebook</button>';
    }
}


/**
 * Fires Before Tutor Student Registration Form
 * @Hook: tutor_load_template_before
 * @Template: dashboard.registration
 */

add_action('tutor_load_template_before', 'tutor_registration_before', 10, 2);
function tutor_registration_before($template, $variable){
    if ($template === 'dashboard.registration'){
        echo '<button>Login with Facebook</button>';
    }
}

/**
 * Fires After Tutor Student Registration Form 
 * @Hook: tutor_load_template_after
 * @Template: dashboard.registration
 */

add_action('tutor_load_template_after', 'tutor_registration_after', 10, 2);
function tutor_registration_after($template, $variable){
    if ($template === 'dashboard.registration'){
        echo '<button>Login with Facebook</button>';
    }
}


/**
 * Fires Before Tutor Instructor Registration Form
 * @Hook: tutor_load_template_before 
 * @Template: dashboard.instructor.registration
 */

add_action('tutor_load_template_before', 'tutor_instructor_registration_before', 10, 2);
function tutor_instructor_registration_before($template, $variable){
    if ($template === 'dashboard.instructor.registration'){
        echo '<button>Login with Facebook</button>';
    }
}

/**
 * Fires After Tutor Instructor Registration Form
 * @Hook: tutor_load_template_after
 * @Template: dashboard.instructor.registration
 */

add_action('tutor_load_template_after', 'tutor_instructor_registration_after', 10, 2);
function tutor_instructor_registration_after($template, $variable){
    if ($template === 'dashboard.instructor.registration'){
        echo '<button>Login with Facebook</button>';
    }
}

Well structured!

can we add a new field to the login page??