  Throughout this paper, we work over ${\mathbb C}$, and $n$ is an integer such
that $n\geq 2$. For an Enriques surface $E$, let $E^{[n]}$ be the Hilbert
scheme of $n$ points of $E$. By Oguiso and Schr\"oer, $E^{[n]}$ has a
Calabi-Yau manifold $X$ as the universal covering space, $\pi :X\rightarrow
E^{[n]}$ of degree $2$. The purpose of this paper is to investigate a
relationship of the small deformation of $E^{[n]}$ and that of $X$ $({\rm
Theorem}\ 1.1)$, the natural automorphism of $E^{[n]}$ $({\rm Theorem}\,1.2)$,
and count the number of isomorphism classes of the Hilbert schemes of $n$
points of Enriques surfaces which has $X$ as the universal covering space when
we fix one $X$ $({\rm Theorem}\,1.3)$.
