K3 surfaces with an order 50 automorphism

In any characteristic p different from 2 and 5, Kondō gave an example of a K3 surface with a purely non-symplectic automorphism of order 50. The surface was explicitly given as a double plane branched along a smooth sextic curve. In this note we show that, in any characteristic $p\ne 2,5$, a K3 surface with a cyclic action of order 50 is isomorphic to the example of Kondō.