Long story short; I posted in MSE

It is well known that if $G$ is a lie group and $H$ is a closed subgroup of $G$, the inclusion $H \hookrightarrow G$ induces a fiber bundle on the classifying spaces

$$ G/H \rightarrow BH \rightarrow BG$$

I am interested in the cohomological Serre spectral sequence (with coefficients in $\mathbb{Z}/2\mathbb{Z}$ for simplicity) associated to this fibration; namely, the $E_2$ term is given by

$$E_2^{p,q} = H^p(BG;\mathcal{H}^q(G/H))$$

If $G$ is path-connected, $BG$ is simply connected and therefore $\mathcal{H}^q(G/H)$ is thus the usual cohomology $H^q(G/H)$; However, what can I say in the case of $G$ being not path-connected? Is there any example where the twisted coefficients are not trivial? or they are trivial in my setting.

but I hope to get better ideas here; Moreover, I want to add the following:

If I work in the ground now, let's say $G = S^1 \times S^1 \times \mathbb{Z}/2\mathbb{Z}$ and $H = \mathbb{Z}/2\mathbb{Z}$ the right factor of $G$, we have a fibre bundle $$ S^1 \times S^1 \rightarrow\mathbb{R}P^\infty \rightarrow \mathbb{C}P^\infty \times\mathbb{C}P^\infty \times \mathbb{R}P^\infty$$

and an action of $\mathbb{Z}/2\mathbb{Z} = \pi_1(BG)$ on $H^1(S^1\times S^1) $. I know the abstract definition of such action (Following

Lecture notes in Algebraic Topology; J. Davis, P. Kirkfor instance), but I don't know how to make it work in an explicit case .

*As a bouns inquiry; is there any good reference to learn how to compute these actions in the case of non-simply connected spaces. Most references that I have reached rather assume that the action is trivial and barely mention the local coefficient system.*