Topological generator of profinite group

Question

I'm trying to prove that $\pi(\operatorname{Spec}\mathbb{Z}[\frac{1}{2}])$ is topologically generated by its $2$-Sylows (Exercise 6.30 of Lenstra's notes on Galois Theory for Schemes). I've been given the hint (by Lenstra himself) that it is sufficient to show that every finite Galois extension of $\mathbb{Q}$ unramified outside $2$ has even order over $\mathbb{Q}$. I managed to prove that this statement about extension is indeed true, but I cannot see how this implies the "topological" thesis. I tried to show that a $2$-Sylow" is dense by showing it intersects every element of a basis of $\pi(\operatorname{Spec}\mathbb{Z}[\frac{1}{2}])$, but as I'm not really able to explicitly describe the $2$-Sylow, I couldn't manage to conclude.