Suppuose that $A:\mathbb{Z}^{p}\rightarrow\text{GL}(q,\mathbb{Z})$ is an action. Define the acction of $\mathbb{Z}^{p}$ on $\mathbb{R}^{p}$ and $\mathbb{T}^{q}$ by:
$$\ell\cdot x=A(\ell)x$$
$$\ell\cdot \overline{x}=A(\ell)\overline{x}$$
Consider the long exact sequence of cohomology groups associated with the exact sequence of $\mathbb{Z}^{p}-$modules $ 0\rightarrow{}\mathbb{Z}^{q}\xrightarrow{i}\mathbb{R}^{q}\xrightarrow{\pi}\mathbb{T}^{q}\xrightarrow{}0$ :
$$\rightarrow H^{1}(\mathbb{Z}^{p},\mathbb{R}^{q})\rightarrow H_{A}(\mathbb{Z}^{p},\mathbb{T}^{q})\rightarrow H_{A}^{2}(\mathbb{Z}^{p},\mathbb{Z}^{q})\rightarrow H_{A}^{2}(\mathbb{Z}^{p},\mathbb{R}^{q})\rightarrow H_{A}^{2}(\mathbb{Z}^{p},\mathbb{T}^{q})\rightarrow{}...$$
Steven Hurder wrote on the paper "Affine Anosov Actions" that
" As pointed out to the author by J. Lewis" one has the following isomorphism:
$$Ker(i^{*})\cong Tor(H_{A}^{2}(\mathbb{Z}^{p},\mathbb{T}^{q}))$$
I need a reference for the proof of that statment. On the Hurder's paper he does not give any reference. I also read the papers form Lewis that Hurder mentioned on the References of his paper but I can't find such a proof.