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They were not looking to sue for peace, to the contrary, they sought victory over the Allies...
Mmmm, not exactly. IIRC, they were looking to make peace with the West so they could concentrate all their fire on Russia.
Also, I get the distinct impression you're confusing the Battle of Britain (summer of 1940) with the invasion of Normandy (summer of 1944). Apologies if I'm mistaken.