Understanding the exactness of the sequence given by the restriction and inflation in Tate-cohomology of dimension $1$ - Neukirch

Question

I am currently reading Neukirch´s "Class Field Theory", but I have problems understanding a certain part of the proof given for:
Let $A$ be a $G$-module and $g$ a normal subgroup of $G$. Then
$0\to H^1\left(G/g, A^g\right)\overset{\text{inf}}{\to}H^1\left(G, A\right)\overset{\text{res}}{\to}H^1\left(g, A\right)$
is exact. In which $\text{res}$ and $\text{inf}$ denote the restriction and inflation respectively.
The part I am struggling with is showing that $\text{Ke res}\subset \text{Im inf}$. The proof goes as follows:
Let $x\colon G\to A$ be a $1$-cocycle of the $G$-module $A$ that is an elemet of the kernel of $\text{res}$, which means
$x(\tau)=\tau a-a \ \ \ \forall \tau\in g$ and some $a\in A$.
From this construct the $1$-cocycle $x´$ with $x`(\tau)=0 \ \ \forall\tau\in g$ given by $x`=x-\rho$ with $\rho$ being the $1$-coboundry given by $\rho(\sigma)=\sigma a -a \ \ \forall\sigma\in G$. The proof then lists two equations for this $x`$ that allow to construct a $1$-cocycle $y\colon G/g\to A$ that is mapped to $x$ under $\text{inf}$. The equations are:
$x`(\sigma-\tau)\color{red}{=}x`(\sigma)+\sigma x`(\tau)= x`(\sigma)$
$x`(\tau\sigma)=x`(\tau)+\tau x`(\sigma)=\tau x`(\sigma)$
each $\forall \tau\in g$.
The second yields from $x`$ being a $1$-cocycle but it is the second equation I don´t get, speciffically the equal sign marked red. I am pretty sure I am overlooking something fundermental, but I have checked over and over and I don´t get why this is equal.
In the book the proof is found on pages 35 and 36 (Part I 4.7. Theorem). Thank you for helping.