It is a question about group cohomology. Supposing that I have a short exact sequence of $G$-modules $1\rightarrow A_1 \rightarrow A_2\rightarrow A_3\rightarrow 1$, I know that there will be a long exact sequence $$\cdots\rightarrow H^n(G,A_1)\rightarrow H^n(G,A_1)\rightarrow H^n(G,A_1)\rightarrow H^{n+1}(G,A_1)\rightarrow \cdots. (1)$$
If $G$ is got from a group extension $1\rightarrow N \rightarrow G\rightarrow G/N\rightarrow 1$, I can use the spectral sequence technique to get the filtered complex $E^{p,q}_{2,A_i}=H^p(G/N,H^q(N,A_i))$ for each group cohomology $H^{p+q}(G,A_{i})$. My question is: will the long exact sequence in Eq.(1) somehow be reflected on each page of the spectral sequence? By that I mean whether there is a exact sequence of $E^{p,q}_{2,A_1}\rightarrow E^{p,q}_{2,A_2} \rightarrow E^{p,q}_{2,A_3}$, or maybe $E^{p,q}_{\infty,A_1}\rightarrow E^{p,q}_{\infty,A_2} \rightarrow E^{p,q}_{\infty,A_3}$. Or perhaps, I should ask whether the spectral sequence in somehow functorial in the coefficients ($G$-modules)?