Let $G$ be an arbitrary group, and let $\mathrm{Aut}(G)$ be the group of automorphisms of $G$ (with composition of morphisms as multiplication).
I'd like to learn more about the problem of characterizing the conjugacy classes $C_T$ (where the indexing variable $T$ ranges over $\mathrm{Aut}(G)$):
$$C_T = \{U\in \mathrm{Aut}(G) \mid (\;\exists\, S \in \mathrm{Aut}(G) \mid S \;{\scriptstyle \circ} \; T \;{\scriptstyle \circ}\;S^{-1} = U \; )\} $$
of $\mathrm{Aut}(G)$.
I know absolutely nothing about this problem. (In fact, for all I know, the problem is too hard—or too easy—to even bring up in polite company.)
What (keywords, books, papers, authors, etc.) should I search for to learn more about this problem?
There's no particular reason that understanding conjugacy classes in an automorphism group would be any easier or harder than understanding conjugacy classes in any other group. There's also not really any general theory for understanding conjugacy -- you mostly just have to do it one group at a time. In fact, I would say that understanding conjugacy in a group usually has more to do with understanding the group well than with having a good general understanding of conjugacy .
To given an example of the difficulty, suppose that $G$ is a nonabelian free group. Then the conjugacy problem in the automorphism group of $G$ is fairly difficult. (See this question on Math Overflow, which pertains to the slightly different problem of conjugacy in the outer automorphism group of a free group).