Structure of $k$-closures of finite nilpotent permutation groups

Let $G$ be a permutation group of a set $\Omega$ and $k$ be a positive integer. The $k$-closure of $G$ is the greatest (w.r.t. inclusion) subgroup $G^{(k)}$ in $\mathrm{Sym} (\Omega)$ which has the same orbits as has $G$ under the componentwise action on the set $\Omega^k$. It is proved that the $k$-closure of a finite nilpotent group coincides with the direct product of $k$-closures of all of its Sylow subgroups.