Abstract algebra help
I assume you mean Inn(H) is the group of inner automorphisms of H and when you say Aut(H) Inn(H), you really mean Aut(H)=Inn(H) -- the inner automorphisms are automorphisms. Also from your conclusion, you left out the necessary assumption that Z(H) is trivial.
Assume H is a normal subgroup of G, Z(H) is trivial and Aut(H)=Inn(H). To show , it is sufficient to show H and CG(H) are normal, and . Since the centralizer is a normal subgroup and Z(H)=<1>, the only thing to prove is :
Let . Then conjugation of elements of H by x is an automorphism of H. So there exists with for all . That is, for all h in H and so . Thus .