I would like some advice on the following problem:
I have a topological group $G=\langle H,g\rangle$, where $H$ is a subgroup and $h$ is an element. Using a result of classifying spaces $Bi:BH \to BG$ can be taken a fiber bundle with fiber $G/H$, where $i:H \hookrightarrow G$ is the inclusion map. It turns out that in my case I have a $2$-fold covering.
I would like to know the cohomology of $BH$. That's why I am trying to use corollary 12.3 from Milnor-Stasheff: "To any $2$-fold covering $\widetilde{B} \to B$ there is associated an exact sequence of the form $\cdots \to H^{j-1}(B) \stackrel{\smile w_1}{\to} H^{j}(B) \to H^{j}(\widetilde{B}) \to H^{j}(B) \to \cdots$"
I already computed the cohomology of $BH$. But I cannot figure it out how to identify $w_1$ for $BG$. Any suggestions?
Thanks in advance!