Let $$0 \longrightarrow A \xrightarrow{\alpha } B \xrightarrow{\beta} C \longrightarrow 0$$ be an exact sequence of $G$-modules. Passing to the sequence of the $G$-invariant submodules won't conserve exactness and the usefulness of the group $H^1(G,A)$ (the first cohomology group), for what I have been able to see, stands in the fact that the sequence $$0 \longrightarrow A^G \xrightarrow{\alpha '} B^G \xrightarrow{\beta '} C^G \xrightarrow{d} H^1(G,A)$$ is exact at least up to $C^G$.
For each $c\in C$, $d$ is defined by taking an element of $B$, $b$ sent in $c$ (the exactness of the first sequence guarantees surjectivity and it is always possible), the 1-coboundary which sends every $\sigma \in G$ to $\varphi_b=\sigma b-b $ and, since it is easy to show that this object belongs to $Im (\alpha)\cong A $, the 1-cocycle $u_c=\alpha^{-1}\circ \varphi_b \in H^1(G,A). u_c$ is set to be the image of $c$ in respect to $d$, that is, if $d$ is a well-defined function, and I'm having a very hard time proving it. Can someone help me?