Would you prefer to broadcast your Facebook live stream from your desktop rather than your mobile phone? If so, then check out OBS Studio. OBS is very similar to Wirecast, but it is completely free of charge.
To broadcast via your desktop you need a stream key. It is easy to find a stream key for your page (under publishing tools > live video), but there are no publishing tools for live streaming on your personal Facebook profile, or from a group.
I spent ages in forums looking for a solution which did not involve any programming skills. I stumbled upon this article, which included a live streaming button pre-programmed for profiles and groups! As if by magic, I could now live stream right from my desktop into my private group.