In the example below I'm calling "first" and then redirecting to "second". I would expect the header "User-Agent" to show the value "Internal" inside the "second" action but there are no response headers, and the request headers are unchanged.
What am I doing wrong, and how can I create an interaction between two actions where headers are exchanged?
The "fakeController" class:
class FakeController extends Controller
{
    public function firstAction()
    {
        $this->response->setHeader("User-Agent","Internal");
        $this->response->redirect("fake/second");
    }
    public function secondAction()
    {
        $rsHdr = $this->response->getHeaders();
        $rqHdr = $this->request->getHeaders();
    }
}