The outer automorphism group of the Suzuki simple group
${}^2B_2 (2^{2m+1})$, $m \geq 1$ is cyclic of order $2m+1$ and
is generated by a field automorphism $\varphi$ of order $2m+1$.
For any almost-simple group  $S \leq H \leq {\rm Aut}(S)$ with
$S={}^2B_2 (2^{2m+1})$, the group $H/S$ is cyclic.
I would like to know

1. the action of $\varphi$ on the conjugacy classes of the group $S$, and

2. the set of complex character degrees of the group $H$.

Thanks for your help.