The problem with writing groups is they exist for people to promote themselves.
The users in the group aren’t going to read you stuff in most cases. They’re too worried about themselves.
Facebook in general is a bad place for free promotion. I worked to build my author page to 100 people. Yet, less than 10 would see my posts. But if I paid for promotion, I could get 1000s of views.

Why is that? Facebook’s business is advertising. So if you want to get good traction there pull out your wallet.