  In this paper, we prove, as the complex case, a supersingular K3 surface over
a field of odd characteristic has an Enriques involution if and only if there
exists a primitive embedding of the twice of the Enriques lattice into the
Neron-Severi group such that the orthogonal complement of the embedding has no
vector of self-intersection -2 using the Crystalline Torelli theorem. By this
criterion and some lattice calculation, we prove that when the characteristic
of the base field is p=19 or p>23, a superingular K3 surface is an Enriques K3
surface if and only if the Artin invariant is less than 6.
