  In this work we introduce a class of discrete groups containing subgroups of
abstract translations and dilations, respectively. A variety of wavelet systems
can appear as $\pi(\G)\psi$, where $\pi$ is a unitary representation of a
wavelet group and $\G$ is the abstract pseudo-lattice $\G$. We prove a
condition in order that a Parseval frame $\pi(\G)\psi$ can be dilated to an
orthonormal basis of the form $\tau(\G)\Psi$ where $\tau$ is a
super-representation of $\pi$. For a subclass of groups that includes the case
where the translation subgroup is Heisenberg, we show that this condition
always holds, and we cite familiar examples as applications.
