Low-degree permutation rational functions over finite fields

We degree all degree-4 rational functions f(x) in F_q(X) which permute P^1(F_q), and answer two questions of Ferraguti and Micheli about the number of such functions and the number of equivalence classes of such functions up to composing with degree-one rational functions. We also determine all degree-8 rational functions f(X) in F_q(C) which permute P^1(F_q) in case q is sufficiently large, and do the same for degree 32 in case either q is odd or f(X) is a nonsquare. Further, for most other positive integers n