  The elliptic modular surface of level 4 is a complex K3 surface with Picard
number 20. This surface has a model over a number field such that its reduction
modulo 3 yields a surface isomorphic to the Fermat quartic surface in
characteristic 3, which is supersingular. The specialization induces an
embedding of the N\'eron-Severi lattices. Using this embedding, we determine
the automorphism group of this K3 surface over a discrete valuation ring of
mixed characteristic whose residue field is of characteristic 3. The elliptic
modular surface of level 4 has a fixed-point free involution that gives rise to
the Enriques surface of type IV in Nikulin-Kondo-Martin's classification of
Enriques surfaces with finite automorphism group. We investigate the
specialization of this involution to characteristic 3.
