On the normalizer of an iterated wreath product

Given a group $G$ and $n\geq 0$, let $W(G,n)$ be the associated iterated wreath product -- unrestricted when $G$ is infinite -- viewed as a permutation group on $G^n$. We prove that the normalizer of $W(G,n)$ in the symmetric group $S(G^n)$ is equal to $M_n\ltimes W(G,n)$, where $M_n$ is isomorphic to~$\mathrm{Aut}(G)^n$. The action of $\mathrm{Aut}(G)^n$ on $W(G,n)$ is recursively described.