Second cohomology group for trivial group action of free abelian group of rank two on group of integers

This article describes the second cohomology group $$H^2(\mathbb{Z}^2;\mathbb{Z})$$ corresponding to the trivial group action of the free abelian group of rank two ($$\mathbb{Z}^2 = \mathbb{Z} \times \mathbb{Z}$$) on the group of integers ($$\mathbb{Z}$$). The second cohomology group is itself isomorphic to $$\mathbb{Z}$$, the group of integers.

Computation in terms of group cohomology
The cohomology group can be computed as an abstract group using the group cohomology of free abelian groups. The general formula is:

$$H^q(\mathbb{Z}^n;\mathbb{Z}) = \mathbb{Z}^{\binom{n}{q}}$$

In our case, $$n = q = 2$$, so $$\binom{n}{q} = 1$$, and we obtain that the cohomology group is $$\mathbb{Z}$$.

Summary
By a priori considerations, all extension groups must have nilpotency class either one or two, and must have Hirsch length as well as polycyclic breadth equal to 3. The minimum size of generating set must be either 2 or 3.