  Let $S$ be a finitely generated pro-$p$ group. Let $\Emb(S)$ 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 $\Emb(S)$ has finitely
many isomorphism classes. For instance, we give an example where $\Emb(S)$
contains an infinite ascending chain of soluble groups, and on the other hand
show that $\Emb(S)$ contains only finitely many isomorphism classes in the case
that $S$ is just infinite.
