A new family of surfaces with pg = q = 2 and K2 = 6 whose Albanese map has degree 4

We construct a new family of minimal surfaces of general type with pg = q = 2 and K2 = 6, whose Albanese map is a quadruple cover of an abelian surface with polarization of type (1, 3). We also show that this family provides an irreducible component of the moduli space of surfaces with pg = q = 2 and K2 = 6. Finally, we prove that such a component is generically smooth of dimension 4 and that it contains the two-dimensional family of product-quotient examples previously constructed by the first author. The main tools we use are the Fourier-Mukai transform and the Schrödinger representation of the finite Heisenberg group H3.