I was wondering if it is possible to classify the finite groups which have no outer automorphisms?
I am currently only aware of the Symmetric Groups ($n \neq 6$) as an infinite class of examples. If there is no classification I would still appreciate any further example of groups which have no outer automorphisms.
By asking GAP to go through the groups of order (up to 110) I found the following groups with trivial outer automorphism:
1, S2, S3, C5 : C4, S4, C7 : C6, C9 : C6, C11 : C10
(These are how GAP describes these groups when I used the function "StructureDescription" and I believe : indicates some sort of semidirect product. In these cases I think the smaller cyclic group may be isomorphic to the automorphism group of the larger one and the action is the action of automorphisms.) From this data, it looks as though there could be a class of groups $C_n \rtimes Aut(C_n)$ which have no outer automorphisms but I am unsure for exactly which $n$ this will be the case.
[Edit: ran through groups of order 1 .. 255 on GAP: 1, C2, S3, C5 : C4, S4, C7 : C6, C9 : C6, S5, S3 x (C5 : C4), (C3 x C3) : QD16, S3 x S4, C13 : C12, (C2 x C2 x C2) : (C7 : C3), (C2 x C2) : (C9 : C6), S3 x (C7 : C6)]