It is well known that the wavelet transform is related to a representation of the
group. The present paper concerns generalizations via a representation of a closed subgroup
. Under certain conditions such a representation leads to a decomposition of
into a finite number of irreducible representations,
(via the action of the semidirect product
). The article focusses on the case where the stabilizer of a generic point in
is a symmetric noncompact subgroup. In particular, it is proved that the generalized wavelet transform is invertible in this case.