My horse wouldn’t keep his fly mask on either. He rubbed it off within 5 minutes of turning him out.
I took a shoelace, cut it in 1/2, cut small holes in the mesh of the fly mask almost to where it meets the fabric for the ears and finally tied a shoelace in each side. When I put the flymask on I loosely tie the shoelaces together with a standard tie so it acts like a throat latch. The fly mask has stayed on and not been rubbed off.