(covers information from several alternate timelines)
The Gestapo was a ruthless police force which existed on Earth from 1933 to 1945. The Gestapo was the primary internal intelligence agency in Germany during the National Socialist period under dictator Adolf Hitler. It was heavily dominated by, although not technically considered part of, the SS.
The Gestapo (abbreviation for Geheime Staatspolizei, or "secret state police") was the agency of Nazi Germany in charge of surveillance of the population to monitor for dissent, disloyalty, and unrest. The group was known to employ arrest and brutal tactics, such as torture, in order to achieve its means. Gestapo leaders were later considered as "evil psychotic men". (TOS: "Patterns of Force")
The Gestapo also employed medical personnel, with some doctors assigned to the agency. In 2267, while infiltrating the Ekosian Chancellery, James T. Kirk ordered the library computer of the USS Enterprise to outfit Leonard McCoy in the uniform of a "Gestapo doctor". (TOS: "Patterns of Force")
A parallel to the Gestapo was drawn in 2267 when, observing the preferred method of career advancement in the mirror universe, Commander Scott said that ISS Enterprise Security Chief Sulu ran an organization "like the ancient Gestapo." (TOS: "Mirror, Mirror")
In a holoprogram set in France of 1944, a French character (portrayed by Neelix) had become friendly with most Gestapo soldiers in Sainte Claire, who loved his cooking, particularly his strudel. (VOY: "The Killing Game")
In an alternate timeline in 1944 Brooklyn, the Gestapo were seen patrolling the streets and going house-to-house to find and recapture Jonathan Archer. They were also seen shooting random people they thought were either in the Resistance, harboring Archer, or both. (ENT: "Storm Front")
Uniforms and insignia
The original Gestapo officers of Nazi Germany wore plain clothes in the execution of their duties, and were not known to wear SS uniforms, although the recreated Ekosian Gestapo wore standard black uniforms of the Ekosian SS.
A group with the same name and purpose, in violation of the rule of non-interference, was later created on the planet Ekos in the mid-2260s by noted Federation historian John Gill. The Ekosian Gestapo was known to employ "frightening methods", a common tactic of which was to infiltrate Zeon resistance cells with apparent sympathizers who would then turn out to be Nazi spies. (TOS: "Patterns of Force")
The headquarters of the Ekosian Gestapo was the Gestapo Command Headquarters. Member's of the Führer's Special Documentary Corps wore uniforms, and were possible members of, the Ekosian Gestapo. (TOS: "Patterns of Force")