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.