I will be happy for any help.

I am interested in any reference or answer about the following question. We know that any automorphism of simple Lie type group is direct of inner, diagonal, field or graph automorphism. In general I like to know what is the structure of $C_{Aut(S)}(\sigma)$ where $S$ is simple lie type and $\sigma$ is an automorphism. "This is my major question" Maybe your answer is the following: Case by case is different. There if I be clear, is it possible to discuss about the centralizer of a diagonal or graph automprphism?