Criterion for a group $G$ to be abelian using $Inn(G)$

Question

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 $\mathcal S(G)$

Thank you in advance!