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 (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.

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.