Does anyone know of a way to invite people from one group to an event for another group? For example, we have an Altar Guild dinner event, created in the Altar Guild group. We want to invite staff members to the event. We have simple RSVP set up for the event to count people.

If it is just one group that you are wanting to join the event, then you could create the same event for the other group.  If you are wanting to invite members from many different groups, it would be best to create a Registration Event, set the Visibility to Everyone, and select the option  "Don't Share on Newsfeed."  Once you Save the Registration Event, you can click the 3 dots beside the event, Click Share, Get Shareable Link, and Post this link to any Groups that you would like to have attend the Event.  This will only generate an email to members of those groups, with whom you share the link for the event.  I hope this helps!