The third homology of stem-extensions and Whitehead's quadratic functor

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 $\Gamma(A) \to A\otimes_{\mathbb{z}} A$, $\gamma(a)\mapsto a\otimes a$.