Let $G$ be a compact group. Then there exists a universal principal $G$-bundle $G\rightarrow E_{G}\rightarrow B_{G}$. Let $X$ be a paracompact $G$-space and suppose that $X\rightarrow X_{G}\rightarrow B_{G}$ is fiber bundle (called as the Borel fibration) associated to the universal bundle. Its structure group is $G$.

**Theorem (The cohomology Leray-Serre spectral sequence])** For the Borel
fibration $X\overset{i}{\rightarrow }X_{G}\rightarrow B_{G}$, there is a
first quadrant spectral sequence of algebras $\left\{ E_{r}^{\ast ,\ast
},d_{r}\right\} $, with
$$
E_{2}^{p,q}\cong H^{p}\left( B_{G};\mathcal{H}^{q}\left( X;%
%TCIMACRO{\U{211a} }%
%BeginExpansion
\mathbb{Q}
%EndExpansion
\right) \right)
$$
the cohomology of $B_{G}$ with local coefficients in the cohomology of $X$,
and coverging to $H^{\ast }\left( X_{G};%
%TCIMACRO{\U{211a} }%
%BeginExpansion
\mathbb{Q}
%EndExpansion
\right) $ as an algebra.

Under some conditions, the local coefficients system is simple, that is, $% \mathcal{H}^{q}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is the usual coefficient $H^{\ast }\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $. For example, the local coefficients system is simple if $B_{G}$ is simply connected (for example, the case $G$ is compact connected Lie group)

**My Question:** Suppose that $G$ is any compact connected group (not
Lie group), and $X$ is a path connected $G$-space, then local coefficient
system is simple?

Or under which conditions, the local coefficients system is simple.