“False flag” is a term that derived from naval warfare and originally meant disguising a ship with a neutral flag in order to get closer to an enemy before attacking. This obviously also has analogues in ground warfare where enemy military units might disguise themselves as civilians or friendly units in order to deceive or passed unnoticed.
I think, however, that you are asking about the term in its modern context popular in conspiracy theories which can be defined as “countries that organize attacks on themselves and make the attacks appear to be by enemy nations or terrorists, thus giving the nation that was supposedly attacked a pretext for domestic repression or foreign military aggression.” (1)
There are notable examples of this. I’m fact, arguably the first aggression of World War II was such an action.
“Operation Himmler” was a series of actions undertaken by Nazi Germany on 31 August 1939 in order to build support for the German invasion of Poland.
The operations - of which there were over 20 - were mainly conducted by the SS and SD. A typical incident would involve SS troops in Polish uniforms attacking a German border post, railway post or radio station and leaving behind bodies in both German and Polish uniforms. In reality the bodies were concentration camp prisoners who had been killed by lethal injection, shot for appearances and left behind.
The most notable of these incidents (the Gleiwitz Incident) was discussed at the Nuremberg Trials (2).
In his speech to the Reichstag justifying war, Hitler specifically cited these incidents as justification:
“I can no longer find any willingness on the part of the Polish Government to conduct serious negotiations with us. These proposals for mediation have failed because in the meanwhile there, first of all, came as an answer to the sudden Polish general mobilization, followed by more Polish atrocities. These were again repeated last night. Recently in one night, there were as many as twenty-one frontier incidents: last night there were fourteen, of which three were quite serious. I have, therefore, resolved to speak to Poland in the same language that Poland for months past has used toward us... This night for the first time Polish regular soldiers fired on our own territory. Since 5:45 a. m., we have been returning the fire... I will continue this struggle, no matter against whom, until the safety of the Reich and its rights are secured”
An academic account of this would be in Peter Longerich’s biography of Himmler, which the Washington Post described as “thoughtful and perceptive” but also - fair warning - “stupendously long and almost unimaginably exhausting”.
Sources:
deHaven-Smith, Lance (2013). Conspiracy Theory in America
Erwin Lahousen Nuremberg evidence: https://avalon.law.yale.edu/imt/11-30-45.asp