Generalized conjugation is the action of a group on its underlying set given by (g,x) -> p(g)xg-1, where p is some fixed endomorphism of G. Here we study combinatorial properties of the sizes of the orbits of the preceding action. In particular, we reduce the problem to a simpler case if p has nontrivial kernel, or if it is an inner automorphism, and we give a construction that allows a partial analysis in the general case.
Achar, Pramod N., "Generalized Conjugacy Classes" (1997). Mathematical Sciences Technical Reports (MSTR). 71.