**Set-up:**


Consider the trivial extension, where $p$ is the projection onto the $\mathbb{Z}\_2$ component,$$1\rightarrow N\rightarrow N\times\mathbb{Z}\_2\xrightarrow{p}\mathbb{Z}\_2\rightarrow 1$$
Define actions of $\mathbb{Z}\_2=\{1,T\}$ on a $\mathbb{Z}\_2$-module $A$ by$$\phi^1\_T:a\mapsto a\ \text{ (trivial)}\ \ \ \text{ and }\ \ \ \phi\_T^2:a\mapsto -a\ \text{ (sign) }$$
Let $N\simeq\ker p$ act trivially on $A$, so that $g\in N\times\mathbb{Z}\_2$ determines maps $a\mapsto\phi^{1,2}\_{p(g)}(a)$.


Now consider the Hochschild-Serre spectral sequence for the group homology, with coefficients in the module $A$, of the extension. (Equivalently, consider the Serre spectral sequence for homology, with coefficients in the local system $A$, of the fibration of Eilenberg-Maclane spaces.) In terms of group homology, the second page is given by
$$E^2\_{p,q}:=H\_p(\mathbb{Z}\_2,H\_q(N;A))\Rightarrow H\_{p+q}(N\times\mathbb{Z}\_2;A)$$
Since the extension is a direct sum, $\mathbb{Z}\_2$ acts trivially on the fiber homology $H\_q(N;A)$.


**Problem:**


It appears that the second page $E^2\_{p,q}$ does not depend on the choice of action (trivial or sign) on the coefficient module while the limit $H\_{p+q}(N\times\mathbb{Z}\_2;A)$ does depend on this choice.


Am I correct?


If so, the choice of $\phi^{1,2}$ must somehow be encoded in the differentials of the sequence.



spectral sequence with non-trivial action on coefficients
I guess here $\mathbb Z/2$-module means $\mathbb Z[\mathbb Z/2]$-module. So let's take $N=1$, $A=\mathbb Z$. $H\_0(\mathbb Z/2,A)$ is the coinvariant
$A/\mathbb Z/2$ so it is $A$ in the case of the trivial action, $\mathbb Z/2$ in the case of the sign action. So $E\_2$ term depends on the action.


