  We study the birational self-maps of the projective plane over finite fields
that induce permutations on the set of rational points. As a main result, we
prove that no odd permutation arises over a non-prime finite field of
characteristic two, which completes the investigation initiated by Cantat about
which permutations can be realized this way. Main ingredients in our proof
include the invariance of parity under groupoid conjugations by birational
maps, and a list of generators for the group of such maps.
