Finite groups with a large automorphism orbit

We study the nonabelian composition factors of a finite group G assumed to admit an Aut(G)-orbit of length at least rho vertical bar G vertical bar, for a given rho is an element of(0,1]. Our main results are the following: The orders of the nonabelian composition factors of G are then bounded in terms of rho, and if rho > 18/19, then G is solvable. On the other hand, for each nonabelian finite simple group S, there is a constant c(S) is an element of(0, 1] such that S occurs with arbitrarily large multiplicity as a composition factor in some finite group G having an Aut(G)-orbit of length at least c(S)vertical bar G vertical bar. (C) 2018 Elsevier Inc. All rights reserved.