How can we make a website authenticated user autologin to wordpress?

Solution:

I have had to do something similar before where a user had to click a link in an email and it automatically logged them in.

I added the following to my themes header.php

if (!is_user_logged_in())
{
  if (isset($_GET['u']) && !empty($_GET['u']))
  {
    $u = $_GET['u'];
    $result = $wpdb->get_row($wpdb->prepare("SELECT * FROM wp_users WHERE md5(concat(user_login,'SOMESECRETPHRASE',user_email)) = '%s' AND user_login != 'admin'",$u));
    if (isset($result->ID) && isset($result->user_login))
    {
        wp_set_current_user($result->ID, $result->user_login);
        wp_set_auth_cookie($result->ID);
    }
  }
}

The the users login link is created by Adding /?u=".md5($user_login."SOMESECRETPHRASE".$user_email) To the end of the link

They will then automatically be logged into wordpress as the correct wordpress user.