Are p-Sylow subgroups of a profinite group dense?

Question

I'm trying to show that $\pi(\operatorname{Spec}\mathbb{Z}\left[\frac{1}{2}\right])$ is topologically generated by its 2-Sylow. I already know that the fundamental group of $\operatorname{Spec}\mathbb{Z}\left[\frac{1}{2}\right]$ arises as the Galois group of the maximal field extension $\mathbb{Q}\subset M $ in which the only prime that ramifies is 2. I also know that this group arises as the inverse limit of all finite extension in which only 2 ramifies. I think (but am not sure) that the 2-Sylows of $\pi$ arise as the inverse limit of an inverse system of 2-Sylows, one for each Galois group of the intermediate finite extensions $\mathbb{Q} \subset L \subset M$. Is this correct? And how do I prove that this 2-Sylow is indeed dense (and so a topological generator)?