How do I set authentication headers to log a client in to another website?

I'm currently working on an autologin feature. The idea is that users can sign in to partner apps with one credential. For this we use the authorization header. I have tried a few different ways to try implement this feature, but without much luck.

In short: I need to send an Authorization header and i am expecting a set-cookie response header. All is going well when i'm making this call in postman, but not when making this call like a user would in the browser. I need the client to be logged in

This is what i have tried so far:

$.ajax({
  url: authUrl,
  type: 'get',
  beforeSend: function (xhr) {
    xhr.setRequestHeader('Authorization', 'Bearer ' + jwt);
    xhr.withCredentials = true;
    xhr.crossDomain = true;
  },
  success: function (response, status, xhr) {
    if (xhr.status !== 200) {
      //Some error handling
      return;
    }
    window.open(form.attr('action'));
  });

The code above does not set a cookie and has no Set-Cookie response header. Presumably due to security conciderations.

//With PHP + Guzzle
$jar = new CookieJar();
$client = new Client();
$response = $client->get($qlikSense->getEndpoint(), [
  'headers' => ['Authorization' => "Bearer {$jwt}"],
  'cookies' => $jar,
]);

foreach ($jar->toArray() as $cookie) {
  setcookie($cookie['Name'], $cookie['Value'], 0, $cookie['Path'],$domain, false, false);
}

I tried to make a curl/guzzle call and get the set-cookie header from there. This did not work as well and likly also due to (obvious) security risks.

My final try was the following:

header("Accept: */*");
header("Authorization: Bearer {$jwt}");
header("location: {$domain}");

The code above just said that it was missing the authentication header, but it did redirect.

Any input or pointers would be much appreciated