  We construct a new family of minimal surfaces of general type with $p_g=q=2$
and $K^2=6$, 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 $p_g=q=2$ and
$K^2=6$. Finally, we prove that such a component is generically smooth of
dimension 4 and that it contains the 2-dimensional family of product-quotient
examples previously constructed by the first author. The main tools we use are
the Fourier-Mukai transform and the Schr\"odinger representation of the finite
Heisenberg group $\mathscr{H}_3$.
