Homological Lyndon–Hochschild–Serre spectral sequence for group extensions

Choose connected CW models from [L1]. Build a based map $BG\to BQ$ inducing $G\to Q$ by mapping the one-skeleton to loops representing the quotient images, filling each two-cell because its attaching word maps to the identity in $Q$, and extending over each higher cell because ${\pi }_k(BQ)=0$ for $k>1$. The finite cell extensions are simultaneous over a set of cells; [A1] supplies those choices. Replace this map by its mapping-path fibration $p:E\to BQ$; the total space $E$ is homotopy equivalent to $BG$, and the fiber $F$ is its homotopy fiber.

The map on fundamental groups is the quotient $G\to Q$. Exactness of [F1] gives ${\pi }_0(F)=\{\ast \}$, ${\pi }_1(F)\cong \ker (G\to Q)=N$, and ${\pi }_k(F)=0$ for every $k>1$, because both $BG$ and $BQ$ have zero higher homotopy groups by [L1]. By the external input [F5], $F$ has CW homotopy type. Choose a connected CW model $F^{\mathrm{\prime }}\simeq F$ and identify ${\pi }_1(F^{\mathrm{\prime }})$ with $N$. The uniqueness theorem [F4] gives $F^{\mathrm{\prime }}\simeq BN$, so $H_q(F;\mathbb{Z})\cong H_q(N;\mathbb{Z})$ for every $q$.

A lift $g\in G$ of a loop class $q\in Q={\pi }_1(BQ)$ transports the fiber fundamental group by $n\mapsto gn{g}^{-1}$. Choose a based homotopy equivalence $h:F^{\mathrm{\prime }}\to F$ as in step 2.1 and a based homotopy inverse $\bar{h}:F\to F^{\mathrm{\prime }}$, using the basepoint path supplied by the transport. The map $T_g^{\mathrm{\prime }}=\bar{h}\circ T_g\circ h$ is a based self-map of $F^{\mathrm{\prime }}$ inducing $c_g(n)=gn{g}^{-1}$ on ${\pi }_1(F^{\mathrm{\prime }})\cong N$. By [F6], the classifying map $c:F^{\mathrm{\prime }}\to BN$ satisfies $c\circ T_g^{\mathrm{\prime }}\simeq B(c_g)\circ c$, so the monodromy on $H_q(F^{\mathrm{\prime }})$ agrees under $h$ with the bar-classifying-space map induced by $c_g$. If a different lift is chosen, it differs by an element of $N$ and changes this map by an inner automorphism of $N$. Inner conjugation is homotopic to the identity on the bar classifying space: the conjugation functor on the one-object category of $N$ is naturally isomorphic to the identity by the conjugating element. Hence the induced action on $H_q(F;\mathbb{Z})\cong H_q(N;\mathbb{Z})$ depends only on $q$, and is exactly the stated $Q$-module action.

Apply [F2] to $p:E\to BQ$. By step 2.1 the fiber homology is $H_q(N;\mathbb{Z})$, and step 3.1 identifies its monodromy local system with the conjugation action. Therefore the $E^2$ page is $H_p(Q;H_q(N;\mathbb{Z}))$ and the spectral sequence strongly converges to $H_{p+q}(E;\mathbb{Z})$.

The total-space equivalence from step 1.1 identifies $H_{p+q}(E;\mathbb{Z})$ with $H_{p+q}(BG;\mathbb{Z})$. By [F4], this classifying CW model is homotopy equivalent to the bar model used in [F3], so its homology is $H_{p+q}(G;\mathbb{Z})$. The base and fiber are connected CW models, so the Serre filtration is first quadrant and finite on each total-degree diagonal. AC supplies the DC hypothesis of [F3] by [L2]. The fiber action and Serre filtration introduce no further choice.