We have the group extension $1 \to F_2 \to \text{SL}(2,\mathbb{Z}) \to \mathbb{Z}_{12} \to 1$, where $F_2$ is the free group on two elements $a$ and $b$. Do we know the explicit expression of
- the corresponding action $\mathbb{Z}_{12} \to \text{Aut}(F_2)$, and
- the corresponding 2-cocycle $\mathbb{Z}_{12} \times \mathbb{Z}_{12} \to F_2$ ?
Also, how can one express a matrix $M \in SL(2,\mathbb{Z})$ as a pair $(x,p)$, with $x \in F_2$, $p \in \mathbb{Z}_{12}$ ? (I know that $\begin{pmatrix}2 & 1\\ 1 & 1\end{pmatrix}$ will be sent to $(a,0)$ and $\begin{pmatrix}1 & 1\\ 1 & 2\end{pmatrix}$ to $(b,0)$ since they are generators of the commutator subgroup, but I'm stuck in the general case)
Any details or references is appreciated.