Hyperelliptic curves of genus 3 with prescribed automorphism group

We study genus 3 hyperelliptic curves which have an extra involution. The locus ${\text{L}}_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli \H_3. We find a birational parametrization of this locus by affine 3-space. For every moduli point \p \in \H_3 such that $|\Aut (\p)|>2$, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points \p \in \H_3 such that $\mathrm{\mid }\text{Aut}(\text{p})\mathrm{\mid }>4$. This is the first time that such a rational model of these curves appears in the literature.