An unusual family of supersingular curves of genus five in characteristic two

We construct a family of smooth supersingular curves of genus $5$ in characteristic $2$ with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus $5$, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-$2$ curve. We also provide an explicit parametrization of this family.