If you don't have the ability to go live inside one of your groups or if you want to go live as your Page instead of your Profile, you can always share a broadcast into your group. Here's how:
1. Click Create, then Live stream, and schedule a live stream to your Facebook Page.
2. Click the 3 dots and click View on Facebook.
3. On the Facebook Post click Share, then click Share to a Group.
4. Select the Group you want to post to.
You must have posting privileges to be able to post your stream into a group.
5. Add a comment (optional), then click Post!
6. Your live stream post will now be posted in your Facebook Group! When you go live, all of your Group members will be able to see your live stream.