Surfaces with p_g=q=2, K^2=6 and Albanese map of degree 2

We classify minimal surfaces $S$ of general type with $p_g=q=2$ and $K_S^2=6$ whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth, irreducible components ${\mathcal{M}}_{Ia}$, ${\mathcal{M}}_{Ib}$, ${\mathcal{M}}_{II}$ of dimension 4, 4, 3, respectively. The general surface $S$ contains a smooth elliptic curve $Z$ such that $Z^2=-2$, which is contracted by the Albanese map and which is preserved by any first-order deformation.