Is it even possible to integrate a plug in that posts to a 'group'
I'm sure FB only allows posting to fan pages and walls.
That being said, I know you can do 'Canvas' app for NodeBB on FB, but this only integrates with FB notifiactions
On the Browser, where as you'll find most of the time they check FB via mobile. And FB mobile is not compatible with
Canvas or any external FB notifications.
So you are probably going to be disappointed with it's results.
Also, on that note. I've found it susccefull for mobile users, using a push notification app. That notifies users on events.
And when they are notified, it refers them back to your forum. via a webui Browser in the app.
This works great with a Responsive design or a dedicated mobile browser.