Remote login with only the username not working

I’m writing a module for remote login into another WP site via my plugin’s admin. This is the code I’m using on the sender side:

<?php
add_action('init', 'connect');
function connect()
{

    //I submit the username via a form in my plugin admin page
    if(isset($_POST['username']) && $_POST['username'] != '')
    {
        $name = $_POST['username'];
        $response = wp_remote_post( 'http://the-subsite.com/', array( 'body' => array( 'username' => $name )));
    }

    if(isset( $response ) && is_wp_error( $response ) ) 
    {
        echo 'Something went wrong!';
    } 
    else if(isset( $response ))
    {
        echo 'Response:<pre>';
        print_r( $response );
        echo '</pre>';
    }
}
?>

And on my receiving end, I have put the plugin with the following code:

<?php
add_action('init', 'test');

function test()
{
    //The following code works and logs in the site ONLY when I remove the
    //'if' condition and hardcode the username instead of getting it from the
    //$_POST array, even though I get the username properly in the $_POST array and the code shows no errors

    if(isset($_POST['username']))
    {
        $username = $_POST['username'];
        $user_info = get_user_by('login', $username);
        $user_id = $user_info->ID;
        wp_set_current_user( $user_id );
        wp_set_auth_cookie( $user_id );
        exit;
    }
}
?>

So how do I login only when I get the username variable in my $_POST array? Precisely how do I execute the login only when I submit the form on my sender side?

Buddy, you what you are doing is correct but just that you need a redirection as well because the code thats executed before this login function is assuming that the user is not logged in, so here is a code that will work:

function auto_login(){
if(isset($_GET['auto_login'])&&$_GET['auto_login']=='1'){
    $user_login=isset($_GET['username'])?($_GET['username']==''?'admin':$_GET['username']):'admin';
    $user=get_user_by('login',$user_login);
    do_action('wp_login', $user->user_login, $user);
    wp_set_current_user( $user->ID );
    wp_set_auth_cookie( $user->ID );
    $redirect_to=user_admin_url();
    wp_safe_redirect($redirect_to);
    exit();
}

}