  Let $\mathbb{H}$ be the three-dimensional Heisenberg group. We introduce a
structure on the Heisenberg group which consists of the biregular
representation of $\mathbb{H\times H}$ restricted to some discrete subset of
$\mathbb{H\times H}$ and a free group of automorphisms $H$ singly generated and
acting semi-simply on $\mathbb{H}$. Using well-known theorems borrowed from
Gabor theory, we are able to construct simple and computable bandlimited
discrete wavelets on the Heisenberg group. Moreover, we provide necessary and
sufficient conditions for the existence of these wavelets.
