In this paper we study the diagonal cosets of the non-compact H₄ WZW model. Generalising earlier work by Antoniadis and Obers, we provide an exact world-sheet description for several families of non-maximally symmetric gravitational plane waves with background NS fluxes. We show that the σ-models that correspond to an asymmetric action of the gauge group smoothly interpolate between singular and non-singular plane waves. We also analyse the representations of the coset chiral algebra and derive the spectrum of all the models.