For context, let $G$ be a compact Lie group and let $H^k(B_G,\mathcal{H}^l(X,K))$ be the cohomology with local coeficcients (twisted by the induced action of $G$ on $H^*(X,K)$) of the classigying space, appearing as the $E_2$-term of the multiplicative Leray-Serre spectral sequence associated to the Borel fibration $X\hookrightarrow X_G\to B_G$ of a free $G$-space $X$. When the induced action of $G$ on $H^*(X)$ is not trivial, we can calculate the additive structure by using the isomorphism $H^k(B_G,\mathcal{H}^*(X,K)\cong Ext^k_G(\mathbb{Z},H^*(X))$ via a projective resolution obtained from the universal cover $E_G$ (as described in Whitehead - Elements of Homotopy Theory, p. 281). By example, for $G=K=\mathbb{Z}_2$, Sun and Wang (https://arxiv.org/abs/2207.14527, p. 8) have done the following computation generalizing an example in Kirk's AT lectures:
Recall that $B_{G}=\mathbb{R}P^\infty $ is a connected CW-complex with one cell in each dimension,
$$
\mathbb{R}P^\infty =e^0\cup e^1 \cup e^2 \cup \cdots,
$$
$E_{G}=S^\infty$ is the universal covering space of $\mathbb{R}P^\infty$, and the corresponding cell decomposition is
$$
S^\infty =e_+^0\cup e_-^0\cup e_+^1\cup e_-^1\cup e_+^2\cup e_-^2\cup \cdots,
$$
with $e_\pm^i$ being the upper and lower hemispheres of the $i$-sphere. According to https://www.maths.ed.ac.uk/~v1ranick/papers/davkir.pdf (Sec. 5.2.1, p. 100), the action of $\pi_1(B_{G})\cong\mathbb{Z}_2$ on $S^{\infty}$ gives $C_*(S^{\infty})$ the structure of a $\mathbb{Z}[\mathbb{Z}_2]$-chain complex, where
$$
\mathbb{Z}[\mathbb{Z}_2]=\mathbb{Z}[g]/\langle g^2-1\rangle=\{a_0+a_1g\mid a_0, a_1\in\mathbb{Z}\}
$$
denotes the group ring. A basis for the free (rank 1) $\mathbb{Z}[\mathbb{Z}_2]$-module $C_i(S^{\infty})$ is $e_+^i$. With the choice of the basis, the $\mathbb{Z}[\mathbb{Z}_2]$-chain complex $C_*(S^{\infty})$ is isomorphic to
$$
\cdots \to\mathbb{Z}[\mathbb{Z}_2] \to \cdots \xrightarrow[]{\,1-g\,} \mathbb{Z}[\mathbb{Z}_2] \xrightarrow[]{\,1+g\,} \mathbb{Z}[\mathbb{Z}_2] \xrightarrow[]{\,1-g\,} \mathbb{Z}[\mathbb{Z}_2] \to 0.
$$
Let
$
\tau =1-g^*,~~\sigma =1+g^*.
$ The cochain complex ${\rm Hom}_{\mathbb{Z}[\mathbb{Z}_2]}(C_*(S^{\infty}),H^l(X;\mathbb{Z}_2))$ is isomorphic to
$$
\cdots \leftarrow H^l(X;\mathbb{Z}_2) \leftarrow \cdots \xleftarrow[]{~\tau~} H^l(X;\mathbb{Z}_2) \xleftarrow[]{~\sigma~} H^l(X;\mathbb{Z}_2) \xleftarrow[]{~\tau~} H^l(X;\mathbb{Z}_2) \leftarrow 0.
$$
So the $E_2$-term of the Leray-Serre spectral sequence associated to the fibration $X\hookrightarrow X_{G}\to B_{G} $ is given by
\begin{align*}
E_2^{k,l} & =H^k(B_{G};\mathcal{H}^l(X;\mathbb{Z}_2))\cong H^k\hskip -1mm\left({\rm Hom}_{\mathbb{Z}[\mathbb{Z}_2]}(C_*(S^{\infty}),H^l(X;\mathbb{Z}_2))\right)\\
& \cong
\begin{cases}
\ker\tau , & k=0,\\
\ker\tau /\text{im }\sigma , & k>0\text{~even},\\
\ker\sigma /\text{im }\tau , & k>0\text{~odd}.
\end{cases}
\end{align*}
Now, my question is about the multiplicative structure of $E_2^{*,*}$ as a bigraded algebra. Assuming the cohomology of $X$ is known as a finitely generated graded-algebra of polynomials, and also the non-trivial action of $G$ on the generators of $H^*(X)$, there is a natural multiplicative structure in $Ext^k_G(\mathbb{Z},H^*(X))$ (maybe induced on the last cochain complex by the cup product in $H^*(X)$) wich matches with the cup product in local coefficients of $H^*(B_{G};\mathcal{H}^l(X;\mathbb{Z}_2)$? Or maybe there are some other methods to obtain the multiplicative structure of a cohomology with local coefficients as graded algebras of polynomias as in the global case? I can't find any literature on it.
Thanks in advance!
