Counterexample (§1, p. 2) — Heisenberg group

There exist a profinite group $G$ whose underlying type lies in universe $0$ and subgroups $N,J,H\le G$ with all the following properties. The group $G$ is finite and, as an abstract group, is isomorphic to $(C_3^{\{0,1,2\}})\rtimes S_3$, where $C_3$ is the additive group of $\mathbb Z/3\mathbb Z$ written multiplicatively, the function group has pointwise multiplication, $S_3$ permutes $\{0,1,2\}$, and the action on functions is $(s\cdot b)(i)=b(s^{-1}(i))$; the semidirect multiplication is $(b,s)(b',s')=(b(s\cdot b'),ss')$. The subgroup $N$ is isomorphic to the group of triples $(a,b,c)\in(\mathbb Z/3\mathbb Z)^3$ with multiplication $(a,b,c)(a',b',c')=(a+a',b+b',c+c'+ab')$, identity $(0,0,0)$, and inverse $(-a,-b,-c+ab)$. The subgroup $J$ is isomorphic to the additive group of $\mathbb Z/6\mathbb Z$ written multiplicatively, and $H$ is isomorphic to $C_3\times S_3$. Their cardinalities are exactly $|G|=162$, $|N|=27$, $|J|=6$, and $|H|=18$. The subgroup $N$ is normal in $G$, every element of $G$ has a unique expression $nj$ with $n\in N,j\in J$, both $N$ and $J$ are nilpotent, and $G$ satisfies the following series condition. For a group $R$, the series condition used here means that there exist $m\in\mathbb N$ and a nondecreasing sequence $(S_i)_{i\in\mathbb N}$ of normal subgroups of $R$ such that $S_0=\{1\}$, $S_m=R$, and, for each $i<m$, some $r_i\in R$ satisfies $S_{i+1}=\langle S_i,r_i\rangle$. Repeated terms are allowed, and $m=0$ is allowed precisely when $R$ is trivial. Each of $N,J,H$ is closed in $G$. For every natural prime $p$ there exist a Sylow pro-$p$ subgroup $P_p$ of $J$ and an element $g_p\in G$ with $g_pP_pg_p^{-1}\le H$, yet there is no $g\in G$ with $gJg^{-1}\le H$, and $N\cap H$ is not normal as a subgroup of $N$. A Sylow pro-$p$ subgroup $P$ of a subgroup $A\le G$ means a subgroup $P\le A$ that is closed in $G$, for which every quotient $P/U$ by an open normal subgroup of $P$ has the property that every element is killed by some power $p^k$ with $k\in\mathbb N$, and that is maximal under inclusion among the closed subgroups of $G$ contained in $A$ with this quotient property. The local witnesses may depend on the prime, and the prime quantifier includes primes other than $2$ and $3$, with trivial Sylow subgroups of $J$. All isomorphisms asserted are group isomorphisms, with no specified compatibility among them. The explicit finite cardinalities exclude trivial groups in this existential assertion, and none of these cardinalities uses the infinite-cardinality value $0$.