The main aim of this paper is to compute the character table
of $\mathrm{G}_{2}(3^{2n+1})\rtimes\langle \sigma \rangle$,
where $\sigma$ is the graph automorphism of $\mathrm{G}_{2}(3^{2n+1})$
such that the fixed-point subgroup $\mathrm{G}_{2}(3^{2n+1})^{\sigma}$
is the Ree group of type $\mathrm{G}_{2}$. As a consequence
we explicitly construct a perfect isometry between the principal
$p$-blocks of $\mathrm{G}_{2}(3^{2n+1})^{\sigma}$ and
$\mathrm{G}_{2}(3^{2n+1})\rtimes\langle \sigma \rangle$ for prime numbers dividing $q^2-q+1$.