The Air Ministry Constabulary was one of the fore runners of the Ministry of Defence Constabulary. It was itself created from the previous Air Ministry Wardens and was formed in 1927/28. The AMC became the Air Force Department Constabulary in 1964, on the Creation of The Ministry of Defence and,finally the MOD Police in 1971, when it amalgamated with the War Department Constabulary and the Admiralty Constabulary. The history of the force can be seen at http://freespace.virgin.net/ian.burdon/force_history.htm
The RAF Police Auxiliaries would seem to be locally recruited at overseas bases. I have a hat which was used in Singapore and I believe they were also used in Cyprus. To dat I have been unable to find any other details about them. If anyone can shed light on this service do please let us know.