'Can the test be completed on a group share aircraft, or does it need to be one with the right type of cert (public transport?) to receive paid flight instruction?'
I think it needs to be on a CofA aircraft. Hence why people who fly Hunters, JPs and Permit Pitts need to visit the cockpit of a C152 for an hour of punishment to revalidate/renew. I think the group ac works if it has a cofa, not sure about public transport- anyone know.
And why can you fulfill the flying requirement (12 hours flying inc 6 pic inc 1 with an instructor) 11 months 25 days before it runs out yet have to wait another 9 months to remember to have it signed in your licence.

Whats the reason behind that gem?