Let $X$ be a nice space with basepoint $x_0 \in X$ and universal covering $\pi \colon \widetilde{X} \to X$.
For a left $\mathbb{Z}[\pi_1(X,x_0)]$-module $V$, the homology of $X$ twisted by $V$ is $$ H^{tw}_n(X;V) = H_n\left(C_*(\widetilde{X}) \otimes_{\mathbb{Z}[\pi_1(X,x_0)]} V\right), $$ where $C_*(\widetilde{X})$ is given the right $\mathbb{Z}[\pi_1(X,x_0)]$-module structure via action $\widetilde{X} \curvearrowleft \pi_1(X,x_0)$.
How do I prove the following Lemma from Exercise 75 of Lecture notes in Algebraic Topology by Davis-Kirk, and what does the notation for the group ring mean when $U$ is not normal?
Shapiro's Lemma: For each subgroup $U$ of $\pi_1(X,x_0)$, there is a (natural?) isomorphism $$ H_*^{tw}\bigl(X;\mathbb{Z}\bigl[\pi_1(X,x_0)/U\bigr]\bigr) \xrightarrow{\cong} H_*(\widetilde{X}/U). $$