A Group G is abelian $\Leftrightarrow$ $ Inn(G)$ is a normal subgroup of Sym(G)

Question

First of all I don´t think that this question is answered here

If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$

because in my opinion this is only one direction of the proof. I would be very thankfull if somebody can do me a favor and answers my question. The question is:

Let (G, *) be a group. Let $\mathcal S(G)=\{f:G\to G\space|f\space is\space bijective\}$ be the symmetric group and $Inn(G)=\{\kappa_{a}\space|\space a\in G\}$ the inner automorphisms. Now I want to show:

G is abelian $\Leftrightarrow$ Inn(G) is a normal subgroup of $\mathcal S(G)$