Odometer actions of the Heisenberg group

Let H₃(?) be the 3-dimensional real Heisenberg group. Given a family of lattices Γ₁ ? Γ₂ ? … ? H₃(?), let T be the associated uniquely ergodic H₃(?)-odometer, i.e., the inverse limit of the H₃(?)-actions by rotations on the homogeneous spaces H₃(?)/Γ_j, j ∈ ?. We describe explicitly the decomposition of the underlying Koopman unitary representation of H₃(?) into a countable direct sum of irreducible components and find the ergodic 2-fold self-joinings of T. We show that in general, the H₃(?)-odometers are neither isospectral nor spectrally determined.