  Let $F$ be a $p$-adic field and $\pi$ be an irreducible smooth representation
of $\textrm{Sp}_{2n}(F)$. In this paper, we show that if $\pi$ and $\pi^\kappa$
are both generic for a common generic character of the maximal unipotent of a
fixed Borel, then $\pi\cong \pi^\kappa$, where $\pi^\kappa$ is the
representation induced by the conjugation action of an element $\kappa\in
\textrm{GSp}_{2n}(F)$. This result is a consequence of the standard local
Langlands conjecture and local Gan-Gross-Prasad conjecture. As a consequence,
we extend the dependence relation of the local Rankin-Selberg gamma factors for
$\textrm{Sp}_{2n}\times \textrm{GL}_m$ on $\psi$ to the general case.
