  We study infinitely iterated wreath products of finite permutation groups
with respect to product actions. In particular, we prove that, for every
non-empty class of finite simple groups $\mathcal{X}$, there exists a finitely
generated hereditarily just infinite profinite group $W$ with composition
factors in $\mathcal{X}$ such that any countably based profinite group with
composition factors in $\mathcal{X}$ can be embedded into $W$. Additionally we
investigate when infinitely iterated wreath products of finite simple groups
with respect to product actions are co-Hopfian or non-co-Hopfian.
