由于重定向规则是静态文本，因此很简单，您只需要popup和declarativeNetRequest API，而不需要内容脚本或后台脚本（service worker）。
manifest.json:

"permissions": ["declarativeNetRequestWithHostAccess"],
  "host_permissions": ["*://*.google.com/"],
  "action": {"default_popup": "popup.html"}

popup.html:

<input id=inp><button id=save>Save</button><div id=err></div>
<script src=popup.js></script>

popup.js:

document.getElementById('save').onclick = () => {
  chrome.declarativeNetRequest.updateDynamicRules({
    removeRuleIds: [1],
    addRules: [{
      id: 1,
      condition: {
        resourceTypes: ['main_frame', 'sub_frame'],
        urlFilter: '||google.com/home/dashboard*',
      },
      action: {
        type: 'redirect',
        redirect: {
          transform: {
            queryTransform: {
              addOrReplaceParams: [{
                key: 'authuser',
                value: document.getElementById('inp').value,
              }],
            },
          },
        },
      },
    }],
  }).catch(e => {
    document.getElementById('err').textContent = e.message;
  });
};