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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 thecorresponding 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 bythe Albanese map and which is preserved by any first-order deformation.

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