The Gun is a light weapon used in Problem Sleuth.
Its innocuous double is the Key. It is the first of the innocuous doubles found in the adventure.
The item first appears as a Gun in Problem Sleuth's Office, with a single bullet in it. Problem Sleuth uses the gun to unlock his door by shooting the lock off. After this point it is not used again until after Problem Sleuth refills it from the Munitions Pack, where it gets handed off to Pickle Inspector, who uses it to shoot Ace Dick by accident. During the battle with Demonhead Mobster Kingpin, Problem Sleuth gives it to Pickle Inspector again, who being unable to carry so many weapons at once, gives it to the Hysterical Dame who then uses them to unlock DMK's armor.
The item originally could only be picked up by PS as the Key, because trying to pick it up as a gun simply turned it back into a key. Early in the adventure, PS attempts to shoot the GUN after using its last shot, resulting in the in-game message "Your KEY is out of bullets!"
The GUN/KEY was removed from the game when it was used in conjunction with the Tommy Gun/Ring of Keys and the Hairpin/Machine Gun to unlock DMK's armor. Although it is very likely from the end of the epilogue he has gotten a new one.