Embedding properties of hereditarily just infinite profinite wreath products

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