  Recently, Hazeltine-Liu, and independently Haan-Kim-Kwon, proved a local
converse theorem for $\mathrm{SO}_{2n}(F)$ over a $p$-adic field $F$, which
says that, up to an outer automorphism of $\mathrm{SO}_{2n}(F)$, an irreducible
generic representation of $\mathrm{SO}_{2n}(F)$ is uniquely determined by its
twisted gamma factors by generic representations of $\mathrm{GL}_k(F)$ 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 $\mathrm{SO}_4(F)$, namely, we show that a generic supercuspidal
representation $\pi$ of $\mathrm{SO}_4(F)$ is uniquely determined by its
$\mathrm{GL}_1$, $\mathrm{GL}_2$ twisted local gamma factors and a twisted
exterior square local gamma factor of $\pi$.
