I too have a Harvey Kay-Chuk.
Mine is marked on the barrel "Harvey .224 Kay-Chuk Std" with the registered trademark symbol.
It is a Model 48, SN K3870xx
I bought it out of Shotgun News around 1974. It came with custom dies, a gauge for loaded rounds and sized cases. Several hundred sized and lubricated gas checked cast bullets, two or three hundred loaded rounds and about the same number of empty cases. I think mine marked "Std" indicates it's not the full blown Kay-Chuk round but closer to a shortened .22 Hornet.
Fun gun to shoot although I don't shoot it often.
JLK