local system of coefficients on a fibration of classyfing spaces

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.