On a refined local converse theorem for SO(4)

Recently, Hazeltine-Liu, and independently Haan-Kim-Kwon, proved a local converse theorem for ${\mathrm{S}\mathrm{O}}_{2n}(F)$ over a $p$-adic field $F$, which says that, up to an outer automorphism of ${\mathrm{S}\mathrm{O}}_{2n}(F)$, an irreducible generic representation of ${\mathrm{S}\mathrm{O}}_{2n}(F)$ is uniquely determined by its twisted gamma factors by generic representations of ${\mathrm{G}\mathrm{L}}_k(F)$ for $k=1,\dots ,n$. It is desirable to remove the ``up to an outer automorphism" part in the above theorem using more twisted gamma factors, but this seems a hard problem. In this paper, we provide a solution to this problem for the group ${\mathrm{S}\mathrm{O}}_4(F)$, namely, we show that a generic supercuspidal representation $\pi$ of ${\mathrm{S}\mathrm{O}}_4(F)$ is uniquely determined by its ${\mathrm{G}\mathrm{L}}_1$, ${\mathrm{G}\mathrm{L}}_2$ twisted local gamma factors and a twisted exterior square local gamma factor of $\pi$.