On the normalizer of an iterated wreath product

Given a group $G$ and $n\ge 0$, 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 $M_n⋉W(G,n)$, where $M_n$ is isomorphic to~$\mathrm{A}\mathrm{u}\mathrm{t}(G{)}^n$. The action of $\mathrm{A}\mathrm{u}\mathrm{t}(G{)}^n$ on $W(G,n)$ is recursively described.