The outer automorphism group of the Suzuki simple group ${}^2B_2 (2^{2m+1})$, m≥1 is cyclic of order 2m+1 and is generated by a field automorphism φ of order 2m+1. For any almost simple group S≤H≤Aut(S) with $S={}^2B_2 (2^{2m+1})$, the group H/S is cyclic. Due to this, I want to know

1. The action of φ on the conjugacy classes of the group S , and so

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

Thanks for your help