Klein coverings over hyperelliptic genus 3 curves

We characterize the moduli space of étale Klein coverings (i.e. Galois with deck group ${\mathbb{Z}}_2^2$) of hyperelliptic curves of genus 3. We prove that the Prym map on each component is injective. As an application, we show that the Prym map of étale Klein coverings of genus 3 curves is generically finite.