Klein coverings of genus 2 curves

We investigate the geometry of etale 4 : 1 coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic, depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on P-1 defining the genus 2 curve. The main result of the paper is the fact that in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties.