The closures of wreath products in product action

Let $m$ be a positive integer and let $\Omega$ be a finite set. The $m$-closure of $G\le{\rm Sym} (\Omega)$ is the largest permutation group $G^{(m)}$ on $\Omega$ having the same orbits as $G$ in its induced action on the Cartesian product $\Omega^m$. An exact formula for the $m$-closure of the wreath product in product action is given. As a corollary, a sufficient condition is obtained for this $m$-closure to be included in the wreath product of the $m$-closures of the factors.