Hi Again,
Sorry for the delay in getting to this today. We are launching a new version of the plugins today and it delayed us in getting to your question.
We do not have features to organize events by groups. You could use the Event Categories for this, but as your request here: “each group needs the facility to add an event where people can register/join” we do not have any feature that could do this. It might be possible with a customization, but that is beyond the support we can provide for the most part.
Let me know if you have any follow up questions.
Thanks