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Let $G$ be a finite group, $M$ a reasonable (e.g. a closed manifold) $G$-space. Then there is a fibration $X \to EG \times_G X \to BG$, where $BG$ is the classifying space of $G$ and $EG$ is its universal cover. Consider the Serre spectral sequence of this fibration: $$ E_2^{p,q} = H^p(BG, \mathcal{H}^q(M; \mathbb{Z})) \Longrightarrow H^{p+q}(EG \times_G X; \mathbb{Z}),$$ with $\mathcal{H}^q(M; \mathbb{Z})$ being a system of local coefficients coming from the $G$-action on $M$.

McCleary writes that "the spectral sequence has an induced action of $H^*(BG;\mathbb{Z})$ on its terms", but gives no explanation whatsoever. Could someone please explain and/or give a reference on how does the $H^*(BG;\mathbb{Z})$-module structure on $E_2$ arise?

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I think this has nothing in particular to do with the Borel fibration.

Given any fiber bundle $F \to E \to B$ with connected fiber $F$, one has $H^0(F;\mathbb Z) \cong \mathbb Z$, and the monodromy action induced on this group by the fundamental group $\pi_1(B)$ of the base is trivial. Thus in the cohomological Serre spectral sequence of this bundle,

$$E_2^{p,0} = H^p(B;\mathbb Z),$$

and the sum (or product, depending on your conventions) of these, the "bottom row" $E_2^{\bullet,0}$, is a subalgebra of the $E_2$ term. The bottom rows of later pages, $E_r^{\bullet,0}$, are quotient rings of $E_2^{\bullet,0}$ and subrings of their respective pages $E_r$. These bottom rows act on the larger algebra by multiplication, so by pulling back to $E_2^{\bullet,0}$, so does it.

That is, it is the maps $$H^*(B;\mathbb Z) = E_2^{\bullet,0} \twoheadrightarrow E_r^{\bullet,0} \hookrightarrow E_r $$ that yield (perhaps anti-climactically) the action.

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