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 $R = A[X,Y]$ (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.