F(p)2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Let F be the finite field of order q(2). It is sometimes attributed to Serre that any curve F-covered by the Hermitian curve Hq+1 : y(q+1) = x(q) +x is also F-maximal. For prime numbers q we show that every F-maximal curve X of genus g >= 2 with vertical bar Aut(X) vertical bar > 84(g - 1) is Galois-covered by Hq+1. The hypothesis on vertical bar Aut(X) vertical bar is sharp, since there exists an F-maximal curve X for q = 71 of genus g = 7 with vertical bar Aut(X) vertical bar = 8.4(7 - 1) which is not Galois-covered by the Hermitian curve H-72.