I have seen that a s.e.s of $G$-modules
$$0\longrightarrow A \longrightarrow B\longrightarrow C \longrightarrow 0$$
gives rise to the following long exact sequence:
$$ 0 \rightarrow H^0(G,A) \rightarrow H^0(G,B) \rightarrow H^0(G,C) \rightarrow H^1(G,A) \rightarrow \ldots.$$ Here, we have a conneting morphism $\Delta: H^i(G,C) \rightarrow H^{i+1}(G,A).$ My question is: What does this connecting morphism do with the cocycles and coboundaries? Any help is greatly appreciated!