Difficult to equate France and Iraq because the societies were so different. France does not have the equivalent to the Shia/Sunni split that exists in Iraq, and although the societal divide at the time between left wing (communist) and right wing (fascist) was large it was not as pervasive nor as violent. When looking at Iraq and Afghanistan I think that many of the problems we face in running a COIN or FID campaign is that we do not understand the environment we are in, and there are often inherent contradictions between what we perceive as as the right thing and what the locals perceive as the right thing, as well as what might be in our interests and what the local elite perceives as in its interest.

That is not to say that there are not lessons in how to take over a country as the German practice of (in the west) maintaining government structures and especially the police are relevant.

I will take advice from those better informed out there (M-A Lagrange?) but I believe that the relevant French Government files pertaining to the period 1940-45 have not yet been released and so the definitive account of the role of the French Police, the milice and the resistance has yet to be written.