On a factorization of an iterated wreath product of permutation groups

We show that if each group of permutations $(G_i, M_i)$, $i\in\mathbb{N}$ has a factorization then their infinite iterated wreath product $\mathop{\wr}\limits_{i=1}^{\infty}\!\! G_i$ also has a factorization. We discuss some properties of this factorization and give examples.