I am TDY so my reference books are unavailable, but.
The Marine Corps first issued helmet covers (frog pattern) in 1943.
That pattern remained in use throught the late 50's when the so called Mitchell Pattern entered the system, and then in the 70's with Woodland.
The Marine Corps issued Dig woodland reversable to Dig Desert for the MICH to Det 1 in 2003
The Army did not issue helmet covers in WW2 but did use two types of netting (wide mesh and small mesh).
During the Korean conflict some Army units used burlap (think the old sandbags) over the helmet, but they went to the Mitchell Pattern in the late 50's as well.
I received my first Crye made MultiCam MICH cover circa 2004.
Re the AirFrame camo cover. Cry makes these. As RgrGordo stated in a previous AirFrame thread, it will not fit the Ops-Core helmet.
Crye- and not OpsCore- will make the helmet cover for OpsCore.
The helmet cover cuts down on shine, something some Euros figured out in WW1. The Germans and Japanese had cloth covers as well, though not universally.
Helmet covers will once more be standard soon, and across the board.