  Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$
be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer
$n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this
article we find the $S$-fundamental group scheme and Nori's fundamental group
scheme of $S^n(X)$.
