  Let $A \rightarrowtail G\twoheadrightarrow Q$ be a stem-extension and let
$\rho: A\times G\to G$ be the multiplication map. We show that there is a
natural map $\varphi: H_1(\Sigma_2^\epsilon, {\rm
Tor}_1^{\mathbb{Z}}({}_{2^\infty}A,{}_{2^\infty}A))\to
H_3(G,\mathbb{Z})/\rho_\ast(A \otimes_{\mathbb{Z}} H_2(G,\mathbb{Z}))$ such
that, the image of $\varphi$ coincides with the image of the natural map
$H_3(A,\mathbb{Z})\to H_3(G,\mathbb{Z})/\rho_\ast(A \otimes_{\mathbb{Z}}
H_2(G,\mathbb{Z}))$. An important tool used here is Whitehead's quadratic
functor $\Gamma$. As part of our proof of the main result, we give a precise
homological description of the kernel of the natural map $\Gamma(A) \to
A\otimes_{\mathbb{z}} A$, $\gamma(a)\mapsto a\otimes a$.
