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We consider a semidirect product of two locally compact groups S and T, with S Abelian, denoted by SσT. An action of SσT on S is introduced to make S a homogeneous space of SσT. Then we define a unitary representation from SσT into the unitary group of L2(S) which is our main tool for defining the continuous wavelet transform on L2(S). Also the main properties of the transform are discussed. We prove the Plancherel and inversion formulas and reproducing kernel’s formula for this transform. This is finally specialized to the case of the continuous wavelet transform on L2(Rd).

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