  We give elementary proofs of the following two theorems on automorphisms of a
finite group G: (1) An automorphism of G is inner if and only if it extends to
an automorphism of every finite group containing G. (2) There exists a finite
group, whose outer automorphism group is isomorphic to G. The first theorem was
proved by Pettet using a graph-theoretical construction of Heineken-Liebeck. A
Lie-theoretical proof of the second theorem was sketched by Cornulier in a
MathOverflow post. Our proofs are purely group-theoretical.
