Not enough information to get further with a common name. There are 5 possible plain "William Kay" in WO338 and I do not have a regiment. Leave Book write him as "W Kaye"
1896 Born. Lancashire
1920 Dec 1. Joined ADRIC with service no 1155. Posted to L coy
1921 Jun 16. Injured in Rathcoole Ambush . Later got £300 compensation for wounds. And was awarded Constabulary Medal
1921 Oct 21 to 10 Nov. Under suspension
1922 Jan 20. Discharged on demobilisation of ADRIC