 To see that S6 has an outer automorphism, recall that homomorphisms 
from a group "G" to a symmetric group S"n" are essentially the same as actions
of "G" on a set of "n" elements, and the subgroup fixing a point is then a subgroup of index at most "n" in "G".Conversely if we have a subgroup of index "n" in "G", the action on the cosets gives a transitive action of "G" on "n" points, and therefore a homomorphism to S"n".Before the more mathematically rigorous constructions, it helps to understand a simple construction.