It a WW2 era US helmet liner
My dad (a WWII vet) had an old helmet liner laying around when I was a kid...you can bet that thing got a ton of use in play battles back then.
I never asked it if it was actually his from the service...I would guess he bought it at a Salvation Army store just for laughs. I guess I never even asked dad if he wore a helmet liner in the service...I'd guess some guys liked to use them and
others preferred to just wear their helmet by itself.
My memory is that it was just a solid olive drab type of color and not multicolored like the one shown, but that's going a long ways back for me.
Anyway, it is some kind of helmet liner in the OPer's pic!