How to submit to a basic, external form page using GM_xmlhttpRequest?

I created a basic script for Greasemonkey in order to search for the word test on Stack Overflow.

According to GM_xmlhttpRequest documentation, parameters for the request should be indicated in the data argument.

// ==UserScript==
// @name        Test Search SO
// @namespace   Test Search SO
// @description Test Search SO
// @include     http://YOUR_SERVER.COM/YOUR_PATH/*
// @version     1
// @grant       GM_xmlhttpRequest
// ==/UserScript==

alert("Ready?");

GM_xmlhttpRequest({
  url: "http://stackoverflow.com/search",
  method: "GET",
  data: "q=test",
  onload: function(response) {
    document.getElementsByTagName("html")[0].innerHTML = response.responseText;
    alert("Done.");
  }
});

Unfortunately, the result is just the page http://stackoverflow.com/search like if data was ignored.

How can I fix this, please?

To send data to a form page, you normally must use POST.
To use POST, you also usually must make sure the encoding is as the page expects.

This works:

GM_xmlhttpRequest ( {
    url:        "http://stackoverflow.com/search",
    method:     "POST",
    data:       "q=" + encodeURIComponent ("test"),
    headers:    {
        "Content-Type": "application/x-www-form-urlencoded"
    },
    onload:     function (response) {
        document.getElementsByTagName ("html")[0].innerHTML = response.responseText;
        alert ("Done.");
    }
} );

(Don't forget to URI encode the parameters.)

Alternatively, many such pages will accept GET requests but then you send the data as URL parameters:

GM_xmlhttpRequest ( {
    url:        "http://stackoverflow.com/search?q=" + encodeURIComponent ("test"),
    method:     "GET",
    onload:     function (response) {
        document.getElementsByTagName ("html")[0].innerHTML = response.responseText;
        alert ("Done.");
    }
} );