The Cohen-Macaulay property of invariant rings over ring of integers of a global field

Let A be the ring of integers of global field K. Let G GL₂ (A) be a finite group. Let G act linearly on R = AX, Y (fixing A). Let RG be the ring of invariants. In the equi-characteristic case we prove RG is Cohen-Macaulay. In mixed characteristic case we prove that if for all primes p dividing |G| the Sylow p-subgroup of G has exponent p then RG is Cohen-Macaulay. We prove a similar case if for all primes p dividing |G| the prime p is un-ramified in K.