You can easily pin a post on a Facebook Group to make it stay on top of the page. You can do so on a computer or mobile device.

You can schedule a post on Facebook for a page or group you manage on a desktop or mobile device. It's easy to stay on top of your content calendar by scheduling Facebook posts ahead of time.

How to post anonymously on Facebook Facebook states that you can post anonymously in a Facebook group only if the admins of the group permit it.