Dummit & Foote defines inner automorphism as :
Let $G$ be a group and let $g \in G$. Conjugation by $g$ is called an inner automorphism of $G$.
Later they say:
If $H$ is a normal subgroup of G, conjugation of an element when restricted to H is an automorphism of H, but need not be an inner automorphism of H.
I don't understand why this should be true, inner automorphism is itself defined as conjugation by an element. Is the restriction to $H$ placed on the element carrying out the conjugation, or the elements on which are conjugated? Later an examples is given of the Klein 4-group, as the normal subgroup of $A_4$, and conjugation defined by an 3-cycle. So the restriction is not on the element defining the conjugation
So why isnt the automorphism of H, an inner automorphism?