Let $A$ be a local noetherian domain. Let $M$ be a torsion free $A$-module equipped with an $A$-linear action of a group $G$. Let $\mathfrak{m}$ be the maximal ideal in $A$.
Is the natural map $M^G \otimes_A A/\mathfrak{m} \rightarrow (M/\mathfrak{m})^G$ injective? If not, what is a counterexample?
I already know how to construct examples where the map is not surjective.