Isomorphism with Ext functor Suppose that $M$ is a free, rank $n$ $R$-module and furthermore, suppose a group $G$ acts on $M$ and the $G$-action commutes with the $R$-action. Hence we have a homomorphism $\rho: G \rightarrow \text{GL}_n(R)$. This in turn gives us a $G$-action on $\text{End}_R(M)$ via $g\cdot \phi = \rho(g) \phi \rho(g)^{-1}$. Now is $$\text{Ext}^1_{\mathbb{Z}[G]}(\mathbb{Z}, \text{End}_R(M)) \cong \text{Ext}^1_{R[G]}(M,M)?$$ I know that the sets are in bijection via a rather roundabout argument, but I am wondering if there is a simple group isomorphism hiding here. Also, are there corresponding isomorphisms of higher Ext groups?
 A: First let $A$ be any $\mathbb{Z}[G]$-module, and $M$ and $N$ any $R[G]$-modules.
There is a natural isomorphism of $R$-modules
$$\operatorname{Hom}_{\mathbb{Z}[G]}\bigl(A,\operatorname{Hom}_R(M,N)\bigr))
\cong
\operatorname{Hom}_{R[G]}\bigl(A\otimes_{\mathbb{Z}}M,N\bigr)$$
(where $G$ acts on $\operatorname{Hom}_R(M,N)$ by $(g\cdot\varphi)(m)=g\varphi(g^{-1}m)$ and on $A\otimes_{\mathbb{Z}}M$ by $g\cdot(a\otimes m)=(ga)\otimes(gm)$), given by
$$\theta\mapsto[a\otimes m\mapsto \theta(a)(m)]$$
with inverse
$$\psi\mapsto [a\mapsto[m\mapsto\psi(a\otimes m)]].$$
From now on assume that $A$ is a projective $\mathbb{Z}[G]$-module and $M$ is projective as an $R$-module. Then as functors from $R[G]$-modules to $R$-modules,
$$\operatorname{Hom}_{R[G]}(A\otimes_{\mathbb{Z}}M, -)\cong
\operatorname{Hom}_{\mathbb{Z}[G]}\bigl(A,\operatorname{Hom}_R(M,-)\bigr),$$
which is exact, so $A\otimes_{\mathbb{Z}}M$ is a projective $R[G]$-module.
Applying these functors of $A$ to a projective resolution $P_*$ of the $\mathbb{Z}[G]$-module $\mathbb{Z}$, we get isomorphic chain complexes
$$\operatorname{Hom}_{\mathbb{Z}[G]}\bigl(P_*,\operatorname{Hom}_R(M,N)\bigr))
\cong
\operatorname{Hom}_{R[G]}\bigl(P_*\otimes_{\mathbb{Z}}M,N\bigr).$$
The homology of the first one is $\operatorname{Ext}^*_{\mathbb{Z}[G]}\bigl(\mathbb{Z}, \operatorname{Hom}_R(M,N)\bigr)$, and (since $P_*\otimes_{\mathbb{Z}}M$ is a projective resolution of$M$ as an $R[G]$-module) the homology of the second is $\operatorname{Ext}^*_{R[G]}(M,N)$.
In particular, if $M=N$ this gives natural isomorphisms of $R$-modules
$$\operatorname{Ext}^n_{\mathbb{Z}[G]}\bigl(\mathbb{Z},\operatorname{End}_R(M)\bigr)\cong
\operatorname{Ext}^n_{R[G]}(M,M)$$
for every $n$.
