Proposition 3.2

For every profinite group $G$ in universe $u$ and closed subgroups $N,H,K\le G$, assume that $N$ is a finite normal subgroup of $G$ and pronilpotent, that either $G$ is prosupersolvable or $G/N$ is pronilpotent, and that every $g\in G$ can be expressed as $nh$ with $n\in N,h\in H$ and also as $n'k$ with $n'\in N,k\in K$. Then there exists $g\in G$ with $gHg^{-1}=K$ if and only if, for every natural prime $p$, there exist a Sylow pro-$p$ subgroup $P$ of $H$, a Sylow pro-$p$ subgroup $Q$ of $K$, and $g_p\in G$ with $g_pPg_p^{-1}=Q$. 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. Saying that a topological group $R$ is pronilpotent means that $R/U$ is nilpotent for every open normal subgroup $U$ of $R$, with the subgroup and quotient topologies understood. 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. Saying that $R$ is prosupersolvable means that every quotient $R/U$ by an open normal subgroup satisfies this series condition. The product decompositions are not required to be unique, and the intersections with $N$ are not required to be trivial. The groups $G,H,K$ need not be finite. Trivial groups are allowed, and every prime is quantified, including primes whose Sylow subgroups are trivial; the local choices can depend on $p$.