The author considers continuous wavelet transforms for the semidirect product group of a unimodular normal subgroup \(N\) with a closed subgroup \(H\) of \(\operatorname{Aut}(N)\). An important feature is that \(N\) is not necessarily commutative. A particular example, when \(N\) is the Heisenberg group, is discussed in detail. The reader is advised also to consult an earlier paper by the same author [J. Fourier Anal. Appl. 12, No. 1, 37–52 (2006; Zbl 1090.42022)].