The invariant of PGU(3,q) in the Hermitian function field

Let F = F|K be a function field over an algebraically closed constant field K of positive characteristic p. For a K-automorphism group G of F, the invariant of G is the fixed field FG of G. If F has transendency degree 1 (i.e. F is the function field of an irreducible curve) and FG is rational, then each generator of FG uniquely determines FG and it makes sense to call each of them the invariant of G. In this paper, F is the Hermitian function field K(Hq)=K(x,y) with yq + y − xq + 1 = 0 and q = pr. We determine the invariant of Aut(K(Hq)) \cong PGU(3,q), and discuss some related questions on Galois subcovers of maximal curves over finite fields.