**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.