Ah. I see the problem. You haven’t studied modern European history. I probably need to start at a fairly elementary level with you.
Petain became Premiere of France June 16, 1940. He agreed to a temporary armistice with Germany on June 22, 1940 which left the French government intact, but permitted German occupation of designated territory still considered under French government. (translation available here: http://avalon.law.yale.edu/wwii/frgearm.asp . The French fleet remained under French command subject to remaining in designated ports. On the other hand, all airfields, even in un-occupied France passed to German control.
The situation was modified as German and Italian troops moved into southern France following the poor showing of French troops against the green American troops in North Africa, but the 1940 French government remained intact.
The French government was finally dissolved in December, 1944.
Puppet governments are not governments. The pattern of Roman client states was being followed here.