Characteristic normal subgroups

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 $\displaystyle f:H \mapsto H by f(h) = nhn^{-1} \ \ \ n \in N$ with $\displaystyle f(H) = H$

Now, since N is normal, we have $\displaystyle gNg^{-1} \in N$, so we have $\displaystyle gHg^{-1} = g f(H) g^{-1} = g ( nhn^{-1} ) g^{-1} = (gn)h(gn)^{-1} $

So H is normal in G.

Is this right?