Proposition 4.2

For every profinite group $G$ in universe $u$, every nonempty set $\Omega$ in universe $v$ with a $G$-action, and closed subgroups $N,H\le G$, suppose that $N$ is normal in $G$ and has commutative multiplication, and that every element of $G$ can be expressed as $nh$ with $n\in N,h\in H$. Suppose that the action is transitive, meaning that for every $x,y\in\Omega$ some $g\in G$ satisfies $g\cdot x=y$, and that each stabilizer $G_x=\{g\in G:g\cdot x=x\}$ is closed in $G$. If for every natural prime $p$ there exist a Sylow pro-$p$ subgroup $P_p$ of $H$ and a point $x_p\in\Omega$ fixed by all elements of $P_p$, then there exists a point $x\in\Omega$ fixed by every element of $H$. 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 points and subgroups in the hypothesis may vary with $p$. No topology on $\Omega$ is specified, and there is no separately assumed continuity of its action beyond the closed-stabilizer condition. Empty $\Omega$ is excluded, but singleton $\Omega$, trivial groups, and trivial Sylow subgroups are allowed. All primes are quantified, no finiteness of $\Omega$ or $G$ is required, and the expression $nh$ need not be unique.