The 2-Weierstrass points of genus 3 hyperelliptic curves with extra automorphisms

For each group $G$, $(\mathrm{\mid }G\mathrm{\mid }>2)$ , which acts as a full automorphism group on a genus 3 hyperelliptic curve, we determine the family of curves which have 2-Weierstrass points. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. The 1-dimensional families that we discover have the property that they contain only genus 0 components.