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The answer is a bit more complicated than your question, but please bear with me.
Of the four ships you mention, only one was actually a ship--HMS Glory. Glory was a Colossus class light carrier laid down in August 1942, launched in November 1943, and commissioned in April 1945, sailing after commissioning to join the British Pacific Fleet just before the end of the war against Japan in September 1945. Glory returned from the Pacific to be placed in reserve in 1947, was recommissioned in 1949 and participated in three deployments to Korea between 1951 and 1953. Glory remained in service until 1956, when she was decommissioned, and she was sold for scrap in 1961.
The three other names mentioned are Royal Navy shore bases, traditionally called 'stone frigates" and so named after sailing ships that were long stationed at these locations. After the ships were broken up, their names remained to represent the base in the Royal Navy's shore establishment. This is what I found on Wikipedia:
HMS Pembroke II was an accounting base at Chatham on the River Medway during World War II. The Royal Dockyard at Chatham has been in existence since the sixteenth century, and HMS Pembroke II was part of a larger shore station. I have not been able to discover what an accounting base is.
HMS Saker was a less traditional member of the Royal Navy's shore establishment, being first established in 1941 as the Royal Naval Air Station located at Dartmouth, Nova Scotia. The name was shifted to an accounting base located at an address on Connecticut Avenue in Washington DC. The name then shifts to a third location in Lewiston, Maine, where it resided at the Naval Auxiliary Air Facility Lewiston from 1943-1945. The name HMS Saker was also used to British sailors stationed at Naval Air Station Squantum Massachusetts and Naval Air Station Brunswick Maine.
HMS Ferret was the Royal Navy shore establishment at Londonderry, Northern Ireland between 1940 and 1947.
As for records of your father's service, I can only offer the following.
For HMS Glory's deck logs, please follow this link:
Based upon what I can see, I presume your father was associated with naval aviation in some way. HMS Glory was built at the Harland and Wolff shipyard in Belfast, Northern Ireland, so your father's association with Londonderry-based HMS Ferret makes sense. The assignment to HMS Pembroke II makes sense if HMS Glory operated out of the Chatham yard after her launching, but I cannot find that detail in HMS Glory's history. The HMS Saker establishment was where Royal Naval Fleet Air Arm squadrons formed and trained with the new American aircraft Great Britain received under Lend Lease.
So my guess is that your father assisted with the formation and training of new Fleet Air Arm aircrews for the squadrons that would serve aboard HMS Glory. As for records, I am at a loss for where to look. Any mention in Royal Navy records outside of the traditional ones found in deck logs would be incidental. Where to look in U.S. Navy records is equally puzzling, as which Navy command administered the Royal Navy aviation training program for Lend Lease aircraft? Perhaps my Research Services colleagues have some idea.
I'm sorry I cannot provide any more clues than these.