This is probably so easy that I didn't find any other questions asking this exact question.
Suppose that $1\to A\to B\to C\to 1$ is an exact sequence of $G$-modules.
I can then easily prove that this sequence induces an exact sequence
\begin{equation} 1\to A^{G}\to B^{G}\to C^{G}. \end{equation} I believe it is an introductory exercise in group cohomology to show that this sequence extends to
\begin{equation} 1\to A^{G}\to B^{G}\to C^{G}\to H^{1}(G,A)\to H^{1}(G,B)\to H^{1}(G,C). \end{equation} Can someone give me a tip on how to define the map from $C^{G}$ to $H^{1}(G,A)$?
I see no obvious way of doing that. Actually, Snake Lemma came to mind but the diagram I have doesn't seem to be of the same form of the one in Snake's hypothesis. Thanks.