Definition ($delta$). Let $c\in C^G$. Then $b\in B$ such that $\psi (b)=c$. Then

$$\psi (\sigma b-b)=\sigma c-c=0\forall \sigma \in G$$

so $\sigma b-b=\varphi (a_{\sigma })$ for some $a_{\sigma }\in A$.

Can check ${(a_{\sigma })}_{\sigma \in G}\in Z^1(G,A)$.

Then $\delta (c)=$ class of ${(a_{\sigma })}_{\sigma \in G}$ in $H^1(G,A)$.