  Let $\mathbb{F}$ be the finite field of order $q^2$, $q=p^h$ with $p$ prime.
It is commonly atribute to J.P. Serre the fact that any curve
$\mathbb{F}$-covered by the Hermitian curve $\mathcal{H}_{q+1}:\,
y^{q+1}=x^q+x$ is also $\mathbb{F}$-maximal. Nevertheless, the converse is not
true as the Giulietti-Korchm\'aros example shows provided that $q>8$ and
$h\equiv 0\pmod{3}$. In this paper, we show that if an $\mathbb{F}$-maximal
curve $\mathcal{X}$ of genus $g\geq 2$ where $q=p$ is such that
$|Aut(\mathcal{X})|>84(g-1)$ then $\mathcal{X}$ is Galois-covered by
$\mathcal{H}_{p+1}$. Also, we show that the hypothesis on the order of
$Aut(\mathcal{X})$ is sharp, since there exists an $\mathbb{F}$-maximal curve
$\mathcal{X}$ for $q=71$ of genus $g=7$ with $|Aut(\mathcal{X})|=84(7-1)$ which
is not Galois-covered by the Hermitian curve $\mathcal{H}_{72}$.
