C})$. Here we explain our method for this.

Let $G$ be a group and $A$ an additive abelian group, which we view as a $G$-module with trivial action. Then a short exact sequence of groups $$\label{eq:ses}
 0 \to A \overset{\iota}{\to} H \overset{\pi}{\to} G \to 1,$$ where $\iota$ and $\pi$ are homomorphisms, such that $\iota(A) \subset Z(H)$ gives a central extension $H$ of $G$ by $A$. Let $M(G,A)$ be the set of such sequences. (Note these sequences are often called central extensions, but for our purpose it makes sense to call the middle term $H$ the central extension.) We say two sequences in $M(G,A)$ are equivalent if there is a map $\phi$ that makes this diagram commute: $$\label{eq:cd}
  \raisebox{-0.5\height}{\includegraphics{commd}}
$$ Let ${\tilde}M(G,A)$ be $M(G,A)$ modulo equivalence.

If two sequences in $M(G,A)$ as above are equivalent, then $H \simeq H'$. However the converse is not true. E.g., taking $G\simeq A \simeq C_p$, then $|{\tilde}M(G,A)| = p$ but there are only two isomorphism classes of central extensions of $C_p$ by itself, namely the two abelian groups of order $p^2$.

Let ${\mathrm{Cent}}(G,A)$ be the set of isomorphism classes of central extensions of $G$ by $A$. Then the above discussion shows we have a surjective 