Let $A$ be complete noetherian local ring with maximal ideal $m$ and residue field $A/m$ a finite field (in other words, $A$ is a noetherian compact local ring)

For which $A$ as above is $GL_2(A)$ the Galois group of a Galois extension of $\mathbb Q$ unramified outside a finite set of primes $S=S(A)$?

For example, we know since the work of Serre on the points of torsion of elliptic curves that when $A=\mathbb Z_p$, the answer is yes. On the other hand, the answer should not be always yes: when $A$ has Krull dimension $5$ or more for instance, such a group would define a deformation to $A$ of the representation $G_{\mathbb Q,S} \rightarrow GL_2(A/m)$, which would have to be a quotient of the universal deformation ring of that representation, and these deformation ring are expected to have dimension at most $4$ (see Mazur's article in Galois Groups over A).

Obviously, I would be very very surprised if I received a complete answer to this question, but is it possible to give a conjectural answer, based on the most optimistic conjectures on the inverse Galois problems (e.g. Shafarevich's conjecture that $Gal(\mathbb Q/ \mathbb Q^{ab})$ is a free profinite group of countable rank), or at least, some reasonable guess?

Let $A$ be complete noetherian local ring with maximal ideal $m$ and residue field $A/m$ a finite field (in other words, $A$ is a noetherian compact local ring)

For which $A$ as above is there a subgroup of $GL_2(A)$ containing $SL_2(A)$ which is the Galois group of a Galois extension of $\mathbb Q$ unramified outside a finite set of primes $S=S(A)$?

(The italic part of the question above has been edited: the first, ill-formulated, version of this question asked if $GL_2(A)$ itself was a Galois group over $\mathbb Q$ and has been answered by Will Savin below).

For example, we know since the work of Serre on the points of torsion of elliptic curves that when $A=\mathbb Z_p$, the answer is yes. On the other hand, the answer should not be always yes: when $A$ has Krull dimension $5$ or more for instance, such a group would define a deformation to $A$ of the representation $G_{\mathbb Q,S} \rightarrow GL_2(A/m)$, which would have to be a quotient of the universal deformation ring of that representation, and these deformation ring are expected to have dimension at most $4$ (see Mazur's article in Galois Groups over A).

Obviously, I would be very very surprised if I received a complete answer to this question, but is it possible to give a conjectural answer, based on the most optimistic conjectures on the inverse Galois problems (e.g. Shafarevich's conjecture that $Gal(\mathbb Q/ \mathbb Q^{ab})$ is a free profinite group of countable rank), or at least, some reasonable guess?

Let $A$ be complete noetherian local ring with maximal ideal $m$ and residue field $A/m$ a finite field of characteristic $p$.

For which $A$ as above is $GL_2(A)$ the Galois group of a Galois extension of $\mathbb Q$ unramified outside a finite set of primes $S=S(A)$?

For example, we know since the work of Serre on the points of torsion of elliptic curves that when $A=\mathbb Z_p$, the answer is yes. On the other hand, the answer should not be always yes: when $A$ has Krull dimension $5$ or more for instance, such a group would define a deformation to $A$ of the representation $G_{\mathbb Q,S} \rightarrow GL_2(A/m)$, which would have to be a quotient of the universal deformation ring of that representation, and these deformation ring are expected to have dimension at most $4$ (see Mazur's article in Galois Groups over A).

Obviously, I would be very very surprised if I received a complete answer to this question, but is it possible to give a conjectural answer, based on the most optimistic conjecture on the inverse Galois problems (e.g. Shafarevich's conjecture that $Gal(\mathbb Q/ \mathbb Q^{ab})$ is a free profinite group of countable rank), or at least, some reasonable guess?

Let $A$ be complete noetherian local ring with maximal ideal $m$ and residue field $A/m$ a finite field (in other words, $A$ is a noetherian compact local ring)

For which $A$ as above is $GL_2(A)$ the Galois group of a Galois extension of $\mathbb Q$ unramified outside a finite set of primes $S=S(A)$?

For example, we know since the work of Serre on the points of torsion of elliptic curves that when $A=\mathbb Z_p$, the answer is yes. On the other hand, the answer should not be always yes: when $A$ has Krull dimension $5$ or more for instance, such a group would define a deformation to $A$ of the representation $G_{\mathbb Q,S} \rightarrow GL_2(A/m)$, which would have to be a quotient of the universal deformation ring of that representation, and these deformation ring are expected to have dimension at most $4$ (see Mazur's article in Galois Groups over A).

Obviously, I would be very very surprised if I received a complete answer to this question, but is it possible to give a conjectural answer, based on the most optimistic conjectures on the inverse Galois problems (e.g. Shafarevich's conjecture that $Gal(\mathbb Q/ \mathbb Q^{ab})$ is a free profinite group of countable rank), or at least, some reasonable guess?