US units usually had their own helmet insignia, showing what unit they were a part of. Normal infantry divisions (usually) had their division insignia painted on the front of the helmet, such as the blue and white "ying yang" of the 29th infantry division. Other units such as the Navy beach battalions and MP's also stenciled their insignia on the front of the helmet. Paratroopers however, had their regiment and battalion stenciled on the side of their helmet rather than a division insignia on the front (as seen in Band of Brothers). In the 101st Airborne a soldier would have a Heart, Diamond, Spade or Club (there was also a circle, a square and a triangle among others. But these signified standalone battalions, platoons and companies such as Artillery battalions etc.) on both sides of their helmet to show what regiment they were a part of along with a small horizontal white stripe signifying their battalion. Some soldiers, mostly officers and some NCO's, also had their rank stenciled on the front of their helmet. As for the back of the helmet there was usually a white stripe indicating whether the soldier was a NCO or an officer (vertical stripe for an officer and a horizontal stripe for NCO's, no stripe for enlisted men). The US army rangers painted a horizontal orange diamond with a number on it on the back of their helmets signifying their battalion (as seen in Saving Private Ryan). There are of course many other examples of helmet markings, but i do not know what the H on your helmet might signify. Maybe you could upload a picture of the helmet to help identify the marking?

Do you have photographs or anything like that? Paint on the back of a helmet rather than the front is odd to say the least. Do you know what country it's from if not American?

There's a billion things it could mean, half of which could be meaningless to everybody outside of the soldiers unit.

Headquarters company is kind of likely, but I can't be sure without seeing.