Can group members go to my private website and schedule group appointments during blocks of available time?

When you schedule blocks of available time in your Console calendar, this is for individual clients only to book appointments with you. And you can reserve those available times for members of a specific group to use.

But clients cannot join group appointments in your calendar, if that is what you have in mind. So, you cannot schedule on the coach end a "group appointment" in your Console calendar, then have people sign up for that group appointment. You must assign your contacts to the group yourself, then schedule the group appointment.

(You can also schedule a group appointment, then add more people to your contact list, assign them to the group, and they too will be able to participate in that group appointment as long as the appointment is still in the future. All the contacts in the group will receive the appointment confirmation, reminders, follow-up email and session notes, provided that they are in the group at the moment when these messages are sent to the group for that particular appointment.)