Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic

We prove that any Fourier\u2013Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the set of Fourier\u2013Mukai partners of a canonical cover of a hyperelliptic or Enriques surface over an algebraically closed field of characteristic greater than three is trivial. These results extend earlier results of Bridgeland\u2013Maciocia and Sosna to positive characteristic