Prove that every characteristic subgroup of a normal subgroup of a group G is a normal subgroup of G, and that every characteristic subgroup of a characteristic subgroup of a group G is a characteristic subgroup of G.

Proof so far:

Suppose that N is a normal subgroup of G, and let H be a charcteristic subgroup of N. Find an inner automorphism of H such that with

Now, since N is normal, we have , so we have

So H is normal in G.

Is this right?