I'm trying to understand how to construct the Lyndon-Hochschild-Serre spectral sequence for the cohomology (with integer coefficients) of the central extension $G$ of a group $Q$ by a group $N$, given a representative cocycle of $H^2(Q,N)$ corresponding to such an extension. I will use the example of $\mathbb Z\_4$, which is a nontrivial central extension of $\mathbb Z\_2$ by $\mathbb Z\_2$. I have tried to explain my reasoning below. 


So I start with the exact sequence


$0 \rightarrow \mathbb Z\_2 \overset{i}{\rightarrow} \mathbb Z\_4 \overset{p}{\rightarrow} \mathbb Z\_2 \rightarrow 0$ 


and compute the $E\_2$-page using $E\_2^{p,q} = H^p(\mathbb Z\_2,H^q(\mathbb Z\_2,\mathbb Z))$. In the case of the trivial central extension, where the action of every group on its respective coefficient module is trivial, we get a page which vanishes for $q$ odd and is just $\mathbb Z\_2$, except for $E\_2^{0,0} = \mathbb Z$, when $q$ is even. This sequence stabilizes on the $E\_2$-page giving the results expected by using, say, the Kunneth formula. 


However, this cannot be the right $E\_2$-page for $\mathbb Z\_4$, because the higher cohomology groups would then be too large. Therefore we must have a nontrivial action of $\mathbb Z\_2$ on the coefficient modules $H^q(\mathbb Z\_2,\mathbb Z)$. This is where I am stuck. I have two questions:


1) What is the above action, and is there a systematic way to see it for large $q$?


2)Is there a systematic way to compute differentials in the $E\_2$-page given the maps $i,p$ and a representative cocycle for this extension?



Cohomology of $\mathbb Z_4$ via the Lyndon-Hochschild-Serre spectral sequence
The action of the quotient on the cohomology groups of the normal subgroup is the trivial action, because the normal subgroup is central. (Think of group cohomology as a functor of the group: the conjugation action of $Q$ on $N$ induces the action of $Q$ on $H^\*(N;\mathbb{Z})$.) 


What happens for the spectral sequence that you are thinking of is that there are non-trivial differentials. In this particular example, all the non-trivial differentials are $d\_3$. In the $E\_2$ page you have that $E\_2^{i,j}$ is either zero or order two (except for $E\_2^{0,0}$ which is infinite cyclic), and the pairs $(i,j)$ for which the group has order two are those for which $j=0$ and $i>0$ is even, or $j>0$ is even and $i$ is arbitrary. 


The differential $d\_3$ cancels lots of these order two groups in pairs, leaving groups of order two in $E\_4^{i,j}$ only in the cases when either $j=0$ and $i>0$ is even or when $j=2$ and $i$ is even. There are lots of tricks for computing differentials. In this case the easiest thing it probably to use the ring structure on each $E\_i^{\*,\*}$, together with the known values of $H^k(\mathbb{Z}\_4;\mathbb{Z})$ for $k=0,1,2$. 


