  Let $A$ be the ring of integers of global field $K$. Let $G \subseteq
GL_2(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y]$ (fixing $A$).
Let $R^G$ be the ring of invariants. In the equi-characteristic case we prove
$R^G$ 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
$R^G$ is Cohen-Macaulay. We prove a similar case if for all primes $p$ dividing
$|G|$ the prime $p$ is un-ramified in $K$.
