I don't know what to add. It is what is known as a "mixmaster" produced by overhaul at Rock Island Arsenal (RIA) with a Colt slide on an Ithaca receiver.
Might the FA be FJA for Frank J. Atwood, the chief inspector at Ithaca? Or FK for Frank Krack at Rock Island.
The "goblet" is a known Ithaca factory inspector's mark.
P is the general purpose proof mark for WW II.
The number 3 is kind of like the little slip of paper in the pocket of your new slacks.
There is probably a source indication on the barrel, visible out of the gun.
HS is very common, for High Standard.
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