  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.
