[*a repost from SE due to the lack of response*]


Given a group $G$, let $A$ be a $G$-module and let $N\trianglelefteq G$.
If I understand it correctly, the superscript "G/N" in the third term of the standard [inflation-restriction](https://en.wikipedia.org/wiki/Inflation-restriction_exact_sequence) exact sequence
$$
0\to H^1(G/N,A^N)\to H^1(G,A)\to H^1(N,A)^{G/N}\to H^2(G/N,A^N)\to H^2(G,A)
$$
means the fixed points under the action of $G/N$ on the first cohomology group
$H^1(N,A)$. But how is this action defined? Is there an elementary definition (in terms of cocycles) that does not refer to the Lyndon–Hochschild–Serre spectral sequence?



The term $H^1(N,A)^{G/N}$ in the inflation-restriction exact sequence
You do not need LHS spectral sequence for this action.
The functors $H^\*(N,-)$ are the derived functors of $(-)^N:G\text{-mod}\rightarrow G/N\text{-mod}$,
so they will carry a structure of $G/N$-modules. 
Explicitly, on the level of cocycles, it can be described as follows:
if you have a one cocycle $f:N\rightarrow A$ and $g\in G$, then the action on $f$ will be given by $g\cdot(f)(n) = g\cdot f(g^{-1}ng)$


