Sadly, this fails to work for me on Ubuntu 18.04.2 LTS.

After installing chrome-gnome-shell 10-1 via apt-get, this extension indicates that it cannot find the native host connector on extensions.gnome.org under Firefox Quantum 66.0.3, stating "Although GNOME Shell integration extension is running, native host connector is not detected. Refer documentation for instructions about installing connector."

The sister extension ( https://chrome.google.com/webstore/detail/gnome-shell-integration/gphhapmejobijbbhgpjhcjognlahblep?hl=en also version 10.1) works fine under the official Chromium build 73.0.3683.86.

Both browsers were installed using Ubuntu Software Center.

Happy to change rating if this can be resolved.

This works. However, don't use it if you install a similar one for Chrome/Chromium, and you enable synchronization of GNOME extensions in Chrome/Chromium (as per https://www.omgubuntu.co.uk/2017/04/sync-gnome-shell-extensions-google-chrome). When I used both, then GNOME integration for Firefox uninstalled GNOME extensions that I had installed in Chrome.

OK