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

I have the native connector chrome-gnome-shell installed, the extension works fine on Chrome [currently running 71.0.3578.98 (Official Build) (64-bit)] but fails to detect the host connector on Firefox Developer Edition [currently running 65.0b5 (64-bit)]:

"Although GNOME Shell integration extension is running, native host connector is not detected. Refer documentation for instructions about installing connector."

OK, I did get the connector to work for Debian. Add the Sid repo from Europe--as indicated on the top of the git page. Edit the /etc/apt/sources.list file from a root terminal--or with sudo (Ubuntu)--as indicated. Update the repos, and chrome-gnome-shell finally should be available. Install that then. Doing that will update to the correct python and other needed files, as well. It's my experience that the careful code-gods of Debian should agree--after a good while of cogitation and the tests of time. They'll then likely reject the Sid repo, and Debian repos will take over....