  The construction of generalized continuous wavelet transforms on locally
compact abelian groups $A$ from quasi-regular representations of a semidirect
product group $G = A \rtimes H$ acting on ${\rm L}^2(A)$ requires the existence
of a square-integrable function whose Plancherel transform satisfies
Calder\'on-type resolution of the identity. The question then arises under what
conditions such square-integrable functions exist.
  The existing literature on this subject leaves a gap between sufficient and
necessary criteria. In this paper, we give a characterization in terms of the
natural action of the dilation group $H$ on the character group of $A$. We
first prove that a Calder\'on-type resolution of the identity gives rise to a
decomposition of Plancherel measure of $A$ into measures on the dual orbits,
and then show that the latter property is equivalent to regularity conditions
on the orbit space of the dual action.
  Thus we obtain, for the first time, sharp necessary and sufficient criteria
for the existence of a wavelet inversion formula. As a byproduct and special
case of our results we obtain that discrete series subrepresentations of the
quasiregular representation correspond precisely to dual orbits with positive
Plancherel measure and associated compact stabilizers. Only sufficiency of the
conditions was previously known.
