As mentioned, I'm moving people from my Facebook page to my group. Well, you can't advertise a group directly on FB anyway. It has to be done on a page or a third party site.
If you're using paid promotion, groups can only be grown via pages or some third party ad - as in displaying on your own website. However, the ad company (for the 3rd party thing) can still be Facebook.
This wouldn't just be because of no effort - but rather the cost of communicating to a page/group is simply too high! I mean, you already probably spent a lot on building it, but what's left to for communication?
Always it's smart to study those already successful. Well, for example, look at a successful Patreon - and study their social media pages. See how many followers they have compared to what profit they make.