Consider an extension\begin{equation}1\rightarrow N\rightarrow G\xrightarrow{\rho} K\rightarrow 1\end{equation}Let $K$ act on a $K$-module $A$ by $\phi\_k: a\mapsto k\cdot a$. Define a $G$-action $a\mapsto g\cdot a:=\rho(g)\cdot a$; under this action, $A$ is a $G$-module. Now consider the Hochschild-Serre spectral sequence\begin{equation}E^2\_{p,q}=H\_p(K,H\_q(N,A))\Rightarrow H\_{p+q}(G,A)\end{equation}It would appear that the LHS, the page $E^2$, does not depend on the $K$-action $\phi\_k$, while the RHS, the limit of the sequence, does depend on it. How does one resolve this apparent contradiction?


To be clear, I understand that, in general, $``H\_p"$ is homology with local coefficients: $K$ acts on $H\_q(N,A)$ according to $\rho$. I am comfortable with this.


I am concerned with the action $\phi\_k$ of $K$ on $A$. Since $N\simeq\ker\rho$, $N$ always acts trivially on $A$, and so the effect of $\phi\_k$ is erased on the second page. Is the data of $\phi\_k$ encoded in the differentials?


I am primarily concerned with the case where $K=\mathbb{Z}\_2=\{1,T\}$. I want to detect the difference between the trivial action $\phi^{triv}\_T:a\mapsto a$ and the inversion action $\phi^{inv}\_T:a\mapsto -a$. Presumably, these should yield different spectral sequences since they yield different limits $H\_n(G,A)$.



Hochschild-Serre spectral sequence and non-trivial action on coefficients
The action of $K$ on $H\_q(N;A)$ may be non-trivial, even though $A$ is a trivial $N$-module. This may even happen if the extension is central, so that the conjugation action of $K$ on $N$ is trivial. You have to take into account the coefficient homomorphisms
$$
(\phi\_k)\_\ast : H\_q(N;A)\to H\_q(N;A),\quad k\in K.
$$
Since $A$ is a non-trivial $K$-module, these may be non-trivial. 


This is treated in section III.8 of Ken Brown's "Cohomology of groups" textbook, where he explains how to view group cohomology as a functor of two variables.


