Group Calendar Widget Display on Group Pages or for Group Members Only?

The plugin is fantastic and mostly meets my client's needs. However we'd like to have the widget only display either on the group pages of the group it references or show anywhere but only for group members. That would be very useful as some of our groups are either private or hidden.

Secondarily, if there could be a way to choose what kind of display the widget shows that would also be useful. For example a mini calendar instead of a list; a scrolling list; mini-pins; are just some ideas.