From Gallian's "Contemporary Abstract Algebra", Chapter 6.
Question 45:
Let $G$ be a group and let $g \in G$. If $z \in Z(G)$, show that the inner automorphism induced by $g$ is the same as the inner automorphism induced by $zg$ (that is, that the mappings $\phi_g$ and $\phi_{zg}$ are equal).
Question 47:
Suppose that $g$ and $h$ induce the same inner automorphism of a group $G$. Prove that $h^{-1}g \in Z(G)$
Let $g, h \in G$.
Then if $h^{-1}g \in Z(G)$ then $\phi_g = \phi_{h^{-1}gg} = \phi_{gh^{-1}g}$ by Exercise 45.
Conversely, if $\phi_g = \phi_{gh^{-1}g}$ then $(gh^{-1}g)^{-1}g = g^{-1}h$ is in $Z(G)$ by Exercise 47, and so its inverse $h^{-1}g$.
Is that it?