Quandles associated to Galois covers of arithmetic schemes

Let $X$ be a normal, separated and integral scheme of finite type over $\mathbb{Z}$ and $\mathcal{M}$ a set of closed points of $X$. To a Galois cover $\tilde{X}$ of $X$ unramified over $\mathcal{M}$, we associate a quandle whose underlying set consists of points of $\tilde{X}$ lying over $\mathcal{M}$. As the limit of such quandles over all 'etale Galois covers and all 'etale abelian covers, we define topological quandles $Q(X,\mathcal{M})$ and $Q^{\mathrm{a}\mathrm{b}}(X,\mathcal{M})$, respectively. Then we study the problem of reconstruction. Let $K$ be $\mathbb{Q}$ or a quadratic field, ${\mathcal{O}}_K$ its ring of integers, $X=$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$ $\mathcal{O}$_K\setminus{\mathfrak{p}}$ the complement of a closed point such that ${\pi }_1(X{)}^{\mathrm{a}\mathrm{b}}$ is infinite, and $\mathcal{M}$ a set of maximal ideals with density $1$. Using results from $p$-adic transcendental number theory, we show that $K$, $\mathfrak{p}$ and the projection $\mathcal{M}\to \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathbb{Z}$ can be recovered from the topological quandle $Q(X,\mathcal{M})$ or $Q^{\mathrm{a}\mathrm{b}}(X,\mathcal{M})$.