  Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $X$
be an irreducible smooth projective surface over $k$. Fix an integer $n \geq 1$
and let ${\mathcal{H}{\it ilb}}_X^n$ be the Hilbert scheme parameterizing
effective $0$-cycles of length $n$ on $X$. The aim of the present article is to
find the $S$-fundamental group scheme and Nori's fundamental group scheme of
the Hilbert scheme $\mathcal{H}{\it ilb}_X^n$.
