In the Esquisse, Grothendieck explains how to find the representations of $G_{\mathbb{Q}}$ on Tate modules of Jacobians through its action on $\widehat{SL(2, \mathbb{Z})}$. This has already been discussed in this question: Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$.

In a footnote to this passage, Grothendieck mentions a way to extend this idea $($here, $\Gamma = G_{\mathbb{Q}})$:

In 1981 I began to experiment with this machine in a few specific cases, obtaining various remarkable representations of $\Gamma$ in groups $G(\widehat{\mathbb{Z}})$, where $G$ is a (not necessarily reductive) group scheme over $\mathbb{Z}$, starting from suitable homomorphisms $$SL(2, \mathbb{Z})\rightarrow G_0(\mathbb{Z}),$$ where $G_0$ is a group scheme over $\mathbb{Z}$, and $G$ is constructed as an extension of $G_0$ by a suitable group scheme. In the "tautological" case $G_0=SL(2)_{\mathbb{Z}}$, we find for $G$ a remarkable extension of $GL(2)_{\mathbb{Z}}$ by a torus of dimension 2, with a motivic representation which "covers" those associated to the class fields of the extensions $\mathbb{Q}(i)$ and $\mathbb{Q}(j)$ (as if by chance, the "fields of complex multiplication" of the two "anharmonic" elliptic curves).

I am interested in understanding what this means, even in the "tautological" case. I can phrase the question as follows.

In general, how does one construct a representation (in this context not necessarily linear, e.g. an outer action) of a suitable extension? That is, given an exact sequence of group schemes $$1\rightarrow N\rightarrow G\rightarrow SL(2, \mathbb{Z})\rightarrow 1,$$ can one produce an action of $G_{\mathbb{Q}}$ on $G(\widehat{\mathbb{Z}})$ starting with an action of $G_{\mathbb{Q}}$ on $\widehat{SL(2, \mathbb{Z})}$? Breaking it down:

1. Even in the trivial case of $N=1$, we have this object $SL(2, \mathbb{Z})(\widehat{\mathbb{Z}})$ whose relation to $\widehat{SL(2, \mathbb{Z})}$ is unclear to me. Is it clear that the representation here is essentially the one we started with?

2. Assuming the first issue is resolved, which $N$ will naturally give an action on $G(\widehat{\mathbb{Z}})$? Perhaps those who themselves carry some action of $G_{\mathbb{Q}}$? In any case, it seems that Grothendieck suggests that tori should be included.

3. What does it mean for a motivic representation to "cover" those associated to the class fields of the extensions $\mathbb{Q}(i)$ and $\mathbb{Q}(j)$? It sounds like there is a separate concept of motivic representations of class fields (??) that I am ignorant of, and they can be realized as quotients of this newly constructed representation.

4. Why does Grothendieck say "a remarkable extension of $Gl(2)_{\mathbb{Z}}$" rather than "a remarkable extension of $Sl(2)_{\mathbb{Z}}$"?

Any insight into these questions, or explanations of better questions towards understanding this passage, would be greatly appreciated.

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the Esquisse, Grothendieck explains how to find the representations of $G_{\mathbb{Q}}$ on Tate modules of Jacobians through its action on $\widehat{\SL(2, \mathbb{Z})}$. This has already been discussed in this question: Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$.

In a footnote to this passage, Grothendieck mentions a way to extend this idea $($here, $\Gamma = G_{\mathbb{Q}})$:

In 1981 I began to experiment with this machine in a few specific cases, obtaining various remarkable representations of $\Gamma$ in groups $G(\widehat{\mathbb{Z}})$, where $G$ is a (not necessarily reductive) group scheme over $\mathbb{Z}$, starting from suitable homomorphisms $$\SL(2, \mathbb{Z})\rightarrow G_0(\mathbb{Z}),$$ where $G_0$ is a group scheme over $\mathbb{Z}$, and $G$ is constructed as an extension of $G_0$ by a suitable group scheme. In the "tautological" case $G_0=\SL(2)_{\mathbb{Z}}$, we find for $G$ a remarkable extension of $\GL(2)_{\mathbb{Z}}$ by a torus of dimension 2, with a motivic representation which "covers" those associated to the class fields of the extensions $\mathbb{Q}(i)$ and $\mathbb{Q}(j)$ (as if by chance, the "fields of complex multiplication" of the two "anharmonic" elliptic curves).

I am interested in understanding what this means, even in the "tautological" case. I can phrase the question as follows.

In general, how does one construct a representation (in this context not necessarily linear, e.g. an outer action) of a suitable extension? That is, given an exact sequence of group schemes $$1\rightarrow N\rightarrow G\rightarrow \SL(2, \mathbb{Z})\rightarrow 1,$$ can one produce an action of $G_{\mathbb{Q}}$ on $G(\widehat{\mathbb{Z}})$ starting with an action of $G_{\mathbb{Q}}$ on $\widehat{\SL(2, \mathbb{Z})}$? Breaking it down:

1. Even in the trivial case of $N=1$, we have this object $\SL(2, \mathbb{Z})(\widehat{\mathbb{Z}})$ whose relation to $\widehat{\SL(2, \mathbb{Z})}$ is unclear to me. Is it clear that the representation here is essentially the one we started with?

2. Assuming the first issue is resolved, which $N$ will naturally give an action on $G(\widehat{\mathbb{Z}})$? Perhaps those who themselves carry some action of $G_{\mathbb{Q}}$? In any case, it seems that Grothendieck suggests that tori should be included.

3. What does it mean for a motivic representation to "cover" those associated to the class fields of the extensions $\mathbb{Q}(i)$ and $\mathbb{Q}(j)$? It sounds like there is a separate concept of motivic representations of class fields (??) that I am ignorant of, and they can be realized as quotients of this newly constructed representation.

4. Why does Grothendieck say "a remarkable extension of $\GL(2)_{\mathbb{Z}}$" rather than "a remarkable extension of $\SL(2)_{\mathbb{Z}}$"?

Any insight into these questions, or explanations of better questions towards understanding this passage, would be greatly appreciated.