I am just posting my comments as one answer.
Let $\mathbb{S}^1$ denote the one-dimensional circle Lie group. The classifying space $B\mathbb{S}^1$ is simply connected with integral cohomology ring isomorphic to the polynomial ring $\mathbb{Z}[c_1]$, where $c_1$ is the element of $H^2(B\mathbb{S}^1;\mathbb{Z})$ representing the first Chern class of the associated $\mathbb{C}$-line bundle over $B\mathbb{S}^1$ induced from the universal principal $\mathbb{S}^1$-bundle, $$\pi:E\mathbb{S}^1\to B\mathbb{S}^1.$$
In particular, $R\pi_*\underline{\mathbb{Z}}$ is quasi-isomorphic to a two-term complex of Abelian sheaves concentrated in degrees $0$ and $1$ whose zeroth cohomology sheaf and first cohomology sheaf are both isomorphic to the locally constant sheaf $\underline{\mathbb{Z}}$ on $B\mathbb{S}^1$ (the zeroth cohomology sheaf is canonically isomorphic to this, but the isomorphism of the first cohomology depends on an orientation of $\mathbb{S}^1$). The induced Leray–Serre Spectral Sequence for $\pi$ reduces to a long exact sequence, usually called the Thom–Gysin Sequence of this principal $\mathbb{S}^1$-bundle, for all $n\geq 1$. $$\dotsb \to H^{n-1}(B\mathbb{S}^1;\mathbb{Z}) \xrightarrow{c_1 \cup -} H^{n+1}(B\mathbb{S}^1;\mathbb{Z}) \to H^{n+1}(E\mathbb{S}^1;\mathbb{Z}) \to \dotsb. $$ Of course the total space $E\mathbb{S}^1$ is connected and contractible, so the long exact sequence reduces to the evident isomorphisms for $n=2m+1$ odd. $$\mathbb{Z}\cdot c_1^{m} \xrightarrow{c_1\cup -} \mathbb{Z}\cdot c_1^{m+1}.$$
Anyway, for every CW complex $B$, for every principal $\mathbb{S}^1$-bundle over $B$, $$\rho:E \to B,$$ there exists a continuous function $f_\rho:B\to B\mathbb{S}^1$ (unique up to homotopy) such that the pullback via $f_\rho$ of the universal principal $\mathbb{S}^1$-bundle $E\mathbb{S}^1$ is isomorphic to $E$ as a principal $\mathbb{S}^1$-bundle over $B$. Thus, by the Proper Base Change Theorem in topology, the derived pushforward $R\rho_*\underline{\mathbb{Z}}$ is isomorphic to the pullback of $R\pi_*\underline{\mathbb{Z}}$. This is quasi-isomorphic to a two-term complex concentrated in degrees $0$ and $1$ whose cohomology sheaves are both isomorphic to the locally constant sheaf $\underline{\mathbb{Z}}$ on $B$. The terms in the induced Leray–Serre Spectral Sequence are naturally modules over the cohomology ring $H^*(B;\mathbb{Z})$. Thus, as above, the spectral sequence reduces to $$\dotsb \to H^{n-1}(B;\mathbb{Z}) \xrightarrow{c_1 \cup -} H^{n+1}(B;\mathbb{Z}) \to H^{n+1}(E;\mathbb{Z}) \to \dotsb. $$ Finally, for every Lie group $G$ (including a finite group with the discrete structure), for every continuous action of $G$ on a CW complex $X$, $$\mu:G\times X \to X,$$ for every morphism of Lie groups, $$\lambda:G\to \mathbb{S}^1,$$ set $B$ equal to $X\times^G EG$, and set $E$ equal to $(\mathbb{S}^1\times X)\times^G EG$, where the action of $G$ on $\mathbb{S}^1\times X$ is the diagonal action of $\lambda$ and $\mu$. In this case, the spectral sequence above reduces to $$\dotsb \to H_G^{n-1}(X;\mathbb{Z}) \xrightarrow{c_1 \cup -} H_G^{n+1}(X;\mathbb{Z}) \to H^{n+1}_G(\mathbb{S}^1\times X;\mathbb{Z}) \to \dotsb. $$
