According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking for a reference for that computes $\pi_1(\mathcal{M}_{d}\times \overline{\mathbb{Q}})$. I think that $\pi_1(\mathcal{M}_{d}\times \overline{\mathbb{Q}})$ is the profinite completion of $\tilde{O}(\Lambda_d)$. Here $\Lambda = U^{\oplus 3} \oplus E_8(-1)^{\oplus 2}$ is the K3 lattice and $e, f$ is a basis of the first hyperbolic plane. Then $\Lambda_{d}$ is $(e + df)^\perp$ and $\tilde{O}_{\Lambda_d}$ are the set of isometries of $\Lambda_d$ that extend to an isometry of $\Lambda$ fixing $e + df$.
My reasoning is that is should be enough to compute $\pi_1(\mathcal{M}_{d} \times \mathbb{C})$. Over $\mathbb{C}$ one can form a complex manifold $N_d$ that is a fine moduli space parameterizing triples $(X, \lambda, \varphi)$ with $X$ a complex K3 surface, $\lambda$ a polarization and $\varphi \colon \Lambda \to H^2(X, \mathbb{Z})$ an isometry sending $e + df$ to $c_1(\lambda)$ (in particular $\lambda$ is primitive). There is an action of $\tilde{O}(\Lambda_d)$ on $N_d$ by acting on the marking. I want to say that the 'quotient' of $N_d$ by $\tilde{O}(\Lambda_d)$ is precisely $\mathcal{M}_{d} \times \mathbb{C}$. I want to say this should be enough to determine that the fundamental group of $\mathcal{M}_d^\text{an}$ is $\tilde{O}(\Lambda_d)$ (by perhaps viewing this as a quotient of a contractible period domain). Then I want to use a comparison between \'etale and topological fundamental groups.
So here are my questions. Is there a reference that computes the fundamental group of $\mathcal{M}_d \times \overline{\mathbb{Q}}$ as a Deligne Mumford stack? If not, is what I said above correct?
