Vintage West German Luftwaffe trench coat

Bought it from an old Aussie who said it was an authentic German trench coat with a bullet hole which he had professionally fixed for $US50 way back! On the picture above you can see the L shaped fix in the lower right side from outside and inside! Other than that it is very well preserved vintage piece by Horstbrink. Size 33! Doublebreasted. One of the removable, screw on metal button that go on a shoulder insignia is missing! On the bottom, they are stamped "Extra Fein" 17 mm and capital A.