Regarding Gregory Berhuy's book "An Introduction to Galois Cohomology and its Applications":

The book defined a cohomology sets for non-abelian $G$-groups. Let $A$ be a $G$-group, we define a 1-cocycle as follows: it is a map $$ \alpha : G \to A \quad , \quad \sigma \mapsto \alpha_\sigma$$ such that for all $\alpha \in A $: $\alpha_1 = 1$ (1 is the unit element of each group) and it satisfies the following relation for all $\sigma , \tau \in G$: $$ \alpha_{\sigma \tau} = \alpha_\sigma \sigma \cdot \alpha_\tau \ . $$

In page 52, Berhuy's uses that fact to deduce that for an exact sequence of groups $$ 1 \to A \stackrel{f}{\to} B \stackrel{g}{\to} C \to 1 $$ where $f(A)$ is a central subgroup of $b$, that $$ g( \beta_\sigma (\sigma \cdot \beta_\tau) \beta_{\sigma \tau}^{-1}) = \gamma_\sigma \sigma \cdot \gamma_\tau \gamma_{\sigma \tau}^{-1} = 1 $$ however, in page 53, he writes $$ f(\alpha_{\sigma, \tau}) = \beta_\sigma (\sigma \cdot \beta_\tau) \beta_{\sigma \tau}^{-1} $$ (notice the brackets on $\sigma \cdot \beta_\tau$) but he doesn't say it is equal to 1, why? If we use the 1-cocycle definition (the equation with the $\alpha$'s) then $$\beta_\sigma (\sigma \cdot \beta_\tau) = \beta_{\sigma \tau} $$

Where is the difference and why?