  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.

Group Theory;Geometric Topology