The number of profinite groups with a specified Sylow subgroup

Let S be a finitely generated pro-p group. Let Ep(S) be the class of profinite groups G that have S as a Sylow subgroup, and such that S intersects nontrivially with every nontrivial normal subgroup of G. In this paper, we investigate whether or not there is a bound on G : S for G Ep(S). For instance, we give an example where Ep(S) contains an infinite ascending chain of soluble groups, and on the other hand show that G : S is bounded in the case where S is just infinite.