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A general spectral formulation is presented, in the spirit of the Karlquist approximation, for the field from a gapped head with a permeable underlayer. For an infinite permeability underlayer, of particular interest in perpendicular recording, the fields and readback response are derived for sine-wave and square-wave magnetization, as well as for an isolated transition. The sine-wave response is given in a simple form, allowing interpretation in terms of a reduced pole density from the medium at the underlayer. The square-wave response is given as a Fourier series of sine-wave amplitudes. Approximate analytical functions are constructed for the response to an isolated transition.