  We determine the full automorphism group of two recently constructed families
$\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ of maximal curves over
finite fields. These curves are covers of the Suzuki and Ree curves, and are
analogous to the Giulietti-Korchm\'aros cover of the Hermitian curve. We also
show that $\tilde{\mathcal{S}}_q$ is not Galois covered by the Hermitian curve
maximal over $\mathbb{F}_{q^4}$, and $\tilde{\mathcal{R}}_q$ is not Galois
covered by the Hermitian curve maximal over $\mathbb{F}_{q^6}$. Finally, we
compute the genera of many Galois subcovers of $\tilde{\mathcal{S}}_q$ and
$\tilde{\mathcal{R}}_q$; this provides new genera for maximal curves.
