  The space of deformations of the integer Heisenberg group under the action of
$\textrm{Aut}(H(\mathbb{R}))$ is a homogeneous space for a non-reductive group.
We analyze its structure as a measurable dynamical system and obtain mean and
variance estimates for Heisenberg lattice point counting in measurable subsets
of $\mathbb{R}^3$; in particular, we obtain a random Minkowski-type theorem.
Unlike the Euclidean case, we show there are necessary geometric conditions on
the sets that satisfy effective variance bounds.
