Groups: The Centralizer of a Subgroup is Normal in the Normalizer of the Subgroup

We prove that if G is a group with subgroup H, then C_G(H) is a normal subgroup of N_G(H).

[speedbird]

Very nice proof,
We know that the subgroup H is normal in its normalizer in the parent group G,
but sir, what can one say about the subgroup H and its centralizer in the parent group G?
Is H a subgroup of its centralizer in G or is the centralizer of H in G a subgroup of H
since the center of H, the centralizer of H in H is a subgroup of H.