  We study automorphisms of order four on K3 surfaces. The symplectic ones have
been first studied by Nikulin, they are known to fix six points and their
action on the K3 lattice is unique. In this paper we give a classification of
the purely non-symplectic automorphisms by relating the structure of their
fixed locus to their action on cohomology, in the following cases: the fixed
locus contains a curve of genus g>0; the fixed locus contains at least a curve
and all the curves fixed by the square of the automorphism are rational. We
give partial results in the other cases. Finally, we classify non-symplectic
automorphisms of order four with symplectic square.
