Normalizer Quotients of Symmetric Groups and Inner Holomorphs

We show that every finite group T is isomorphic to a normalizer quotient NSₙ(H)/H for some n and a subgroup H≤ Sₙ. We show that this holds for all large enough n≥ n₀(T) and also with Sₙ replaced by Aₙ. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group G with Out(G) T (for any given finite group T) and the determination of the normalizer in Sym(G) of the inner holomorph InHol(G)(G) for any centerless indecomposable finite group G, which may be of independent interest.