I came across the following problem when I was trying to construct a certain type of homomorphisms from $\Gamma_{\mathbb Q}$ to $E^{sc}_7(\mathbb F_p)$ for any prime $p$:
Is the double cover of $Sp_6(\mathbb F_2)(=PSp_6(\mathbb F_2))$ known as a Galois group over $\mathbb Q$?
A little background about $Sp_6(\mathbb F_2)$: It is a finite simple group of order $2^9\cdot 3^4 \cdot 5 \cdot 7$ with Schur multiplier 2 (sometimes denoted by $C_3(2)$). So it possesses a unique non-trivial two-fold central extension which I shall denote by $\widetilde{Sp}_6(\mathbb F_2)$. It is a quasisimple group. On the other hand, $Sp_6(\mathbb F_2)$ is isomorphic to the derived subgroup of the Weyl group of type $E_7$ (denoted by $W=W(E_7)$), which is isomorphic to $Sp_6(\mathbb F_2)\times \{\pm 1\}$. It follows that $\widetilde{Sp}_6(\mathbb F_2)$ can be realized as the pullback of $Sp_6(\mathbb F_2) \cong [W,W] \to SO(7)$ along the projection $Spin(7) \to SO(7)$. Let $G=E_7^{sc}$ be the simply-connected Chevalley group of type $E_7$. Let $T$ be a maximal torus of $G$. Consider the following exact sequence: $$1\to T \to N_G(T) \to N_G(T)/T\cong W \to 1$$ By a result of J.Adam and X.He, https://arxiv.org/pdf/1608.00510.pdf, the sequence does not split. Moreover, by going along their line of argument, it is not hard to see that the sequence does not split for $[W,W]\cong Sp_6(\mathbb F_2)$ either. I hope to build a homomorphism $\Gamma_{\mathbb Q} \to E_7^{sc}(\mathbb C)$ whose image is isomorphic to the inverse image of $W$ in $N_G(T)$. This problem can be easily reduced to the problem of realizing $\widetilde{Sp}_6(\mathbb F_2)$ as a Galois group over $\mathbb Q$.
It is known that the Weyl group of a reductive group $G$ can be realized a Galois group over $\mathbb Q$: the idea is that the Galois group of the characteristic polynomial of $Ad(g)|_{\mathfrak g}$ for "generic" $g\in G(\mathbb Q)$ is isomorphic to the Weyl group of $G$, provided that $G/\mathbb Q$ is split. This dates back to E.Cartan's famous thesis 120 years ago! See Joulve,Kowalski and Zywina's paper https://people.math.ethz.ch/~kowalski/weyl-group-e8.pdf for a good account of this story. In particular, the finite simple group $Sp_6(\mathbb F_2)$ can be realized as a Galois group over $\mathbb Q$ (being a direct summand of $W(E_7)$). To attack $\widetilde{Sp}_6(\mathbb F_2)$, it is natural to consider the embedding problem posed by the non-split exact sequence: $$1 \to \{\pm 1\} \to \widetilde{Sp}_6(\mathbb F_2) \to Sp_6(\mathbb F_2) \to 1$$ I don't know how to solve this problem. Any suggestions would be appreciated. On the other hand, since this group is a concrete group with generators and relations, I wonder if one can find a polynomial over $\mathbb Q$ that realizes it using computer algorithms.
