Normalizer Quotients of Symmetric Groups and Inner Holomorphs

We show that every finite group $T$ is isomorphic to a normalizer quotient $N_{S_n}(H)\mathrm{/}H$ for some $n$ and a subgroup $H\le S_n$. We show that this holds for all large enough $n\ge n_0(T)$ and also with $S_n$ replaced by $A_n$. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group $G$ with $\mathrm{O}\mathrm{u}\mathrm{t}(G)\cong T$ (for any given finite group $T$) and the determination of the normalizer in $\mathrm{S}\mathrm{y}\mathrm{m}(\mathrm{G})$ of the inner holomorph $\mathrm{I}\mathrm{n}\mathrm{H}\mathrm{o}\mathrm{l}(G)\le \mathrm{S}\mathrm{y}\mathrm{m}(G)$ for any centerless indecomposable finite group $G$, which may be of independent interest.