It is known (given CFSG) that all non-abelian finite simple groups have small outer automorphism groups. However, it's quite tedious to list all the possibilities. Does anyone know a reference for a statement of the following form?


Let G be a non-abelian finite simple group. Then |Out(G)| < f(|G|) (where f is something straightforward that gives a good idea asymptotically of the worst case).


Also, a related question: How good a bound can be obtained in this case without using the classification? Can one do much better than the bounds one might obtain for the outer automorphism group of an arbitrary finite group?



Estimate for the order of the outer automorphism group of a finite simple group
Check article "Probabilistic generation of wreath products of non-abelian finite simple groups" by Martyn Quick. In Section 3.1 he consider this question and get $|Out G|\leq |G|/30$ for every non-abelian finite simple group $G$, which was enough for his needs. 


