  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 \cite{bo} 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 $\mathbb{P}^1$
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.
