  We present a method to generate many automorphisms of a supersingular K3
surface in odd characteristic. As an application, we show that, if p is an odd
prime less than or equal to 7919, then every supersingular K3 surface in
characteristic p has an automorphism whose characteristic polynomial on the
N\'eron-Severi lattice is a Salem polynomial of degree 22. For a supersingular
K3 surface with Artin invariant 10, the same holds for odd primes less than or
equal to 17389.
