In this paper we study automorphisms of order four on K3 surfaces. We give a classification of the non-symplectic ones when either the square of the automorphism is symplectic, or its fixed locus contains a curve of positive genus, or its fixed locus contains at least a curve and all the curves fixed by its square are rational. We provide partial results in the other cases.
"Symmetries of order four on K3 surfaces." J. Math. Soc. Japan 67 (2) 503 - 533, April, 2015. https://doi.org/10.2969/jmsj/06720503