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

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

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

Thanks for your help.

Atlas of Finite Groups. For example, $S=$Sz(32) involves $m=2$, while $\varphi$ has order 5; here theAtlasseems to answer both of your questions, if you unpack the notational conventions. But in general what you are asking for probably gets much longer computationally even though the character degrees for the Suzuki groups are worked out in Suzuki's original papers. (Those papers may also be worth consulting.) $\endgroup$ – Jim Humphreys Oct 11 '15 at 19:34