MP 44/StG44

The MP43/MP44/StG44 the latter being renamed of the former came into design in 1943.The MP43 would replace the MKb.42(H)and the MKb.42(W)assault rifles that were designed in 1942 and put into service in small numbers by 1943. Hitler was finally convinced to give the go ahead for the production of the MP43. The MP43/44 went into production in 1943 and issued to troops on the eastern front on July of 1944. These new assault rifles were to replace the sub machine guns and bolt action rifles. Unfortunally, it was to little to late to save the Third Reich.
The answer to your problem is that the assault rifle that probably appeared in Normandy would have been the MKb.42 since the MP43/44 were nearly identical with small production differences and dates to the MKb.42.
The research into the assault rifle interest me because I am currently working on a project that involves the MP43/44 and so I had the info already at hand. The book I used was valuable to my research. GUNS of the REICH,Firearms of the German Forces,1939-1945 by George Markham.Hope this info helps clarify everything.

Henry