  Let $G$ be a quasi-split tamely ramified connected reductive group defined
over a $p$-adic field $F$. We show that if $-1$ is in the $F$-points of the
absolute Weyl group of $G$, then self-dual supercuspidal representations of
$G(F)$ exist. Now assume further that $G$ is unramified and that the center of
$G$ is connected. Let $\pi$ be a generic self-dual depth-zero regular
supercuspidal representation of $G(F)$. We show that the Frobenius-Schur
indicator of $\pi$ is given by the sign by which a certain distinguished
element of the center of $G(F)$ of order two acts on $\pi$.
