Embedding properties of hereditarily just infinite profinite wreath\n products

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