The Calabi invariant and The Lyndon-Hochschild-Serre spectral sequence

Let $G$ be a group and $N$ be a normal subgroup of $G$. There exists the group extension $G$ of $G/N$ by $N$. For a $G$-module $A$ which $N$ acts on trivially and a $G$-invariant homomorphism on $N$ to $A$, we obtain a central extension of $G/N$ by $A$. By using connection cochains, we exhibit the formula of its extension class such that clarify the relation among connection cochains, extension classes and the LHS spectral sequence.