TITLE: Inverse Galois problem for $GL_2$ of a compact local ring
QUESTION [8 upvotes]: 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?

REPLY [2 votes]: This isn't an answer, just some idle thoughts about the following closely related problem:
Which complete local Noetherian rings $A$ 
can occur as the quotient of the universal deformation ring $R$ of a representation
$\rho: G_{\mathbb{Q},S} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$? 
Such a universal ring $R$ should always have dimension $4$ and  be a complete intersection, at least conjecturally.   
So straight away  the following question occurs:
is the set $\mathcal{X}$ of   isomorphism classes of complete, local, Noetherian, integral,  $4$-dimensional  $W(\mathbb{F}_q)$-algebras actually uncountable?    (Conditions slightly edited)
I don't know the answer  but if indeed $\mathcal{X}$ is uncountable, that
seems to be a problem:  Only finitely many of the elements of $\mathcal{X}$ could occur as quotients of each of the countably many  universal $R$s (namely, one for each minimal prime ideal of $R$).  
On the other hand, I think one  might be able to verify   that
the set of universal deformation rings is dense in $\mathcal{X}$ (for  some reasonable topology).  For example,
I believe that any Artin local  $A$ is a quotient of an $R$ as above. This  should follow from the method of Taylor and Wiles: they show that you can get a formal power series ring, in as many variables as you want, as a limit of such deformation rings $R$.