The number of profinite groups with a specified Sylow subgrou

Let $S$ be a finitely generated pro-$p$ group. Let $\text{Emb}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects non-trivially with every non-trivial normal subgroup of $G$. In this paper, we investigate the question of whether or not $\text{Emb}(S)$ has finitely many isomorphism classes. For instance, we give an example where $\text{Emb}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $\text{Emb}(S)$ contains only finitely many isomorphism classes in the case that $S$ is just infinite.