Consider the two (inequivalent) $\mathbb{Z}$-representations $\phi,\psi$ of the symmetric group $S=S\_3$ given by


$(1,2)^\phi=\left(\begin{array}{rr}0 &-1\\\ -1 & 0\end{array}\right), \qquad
(1,2,3)^\phi=\left(\begin{array}{rr}0 &1\\\ -1 & -1\end{array}\right);$


$(1,2)^\psi=\left(\begin{array}{rr}0 &1\\\ 1 & 0\end{array}\right), \qquad
(1,2,3)^\psi=\left(\begin{array}{rr}0 &1\\\ -1 & -1\end{array}\right).$


Now, let $F=\langle x,y\rangle$ be a free 2-generated group. The representation $\phi$ can be "lifted" to an embedding $\tau:S\to\rm{Aut}(F)$ as follows:


$(1,2)^\tau=[x\mapsto y^{-1};\quad y\mapsto x^{-1}], \qquad
(1,2,3)^\tau=[x\mapsto y;\quad y\mapsto x^{-1}y^{-1}].$


**Question.** Can one similarly lift $\psi$?


**Remark 1.** By "lifting" a representation $\phi:S\to\rm{GL}\_2(\mathbb{Z})$ I mean finding an embedding $\tau:S\to\rm{Aut}(F)$ such that $\phi=\tau\alpha$, where $\alpha:\rm{Aut}(F)\to\rm{GL}\_2(\mathbb{Z})$ is the natural epimorphism.


**Remark 2.** A naïve attempt to send
$(1,2)\ \mapsto\ [x\mapsto y;\quad y\mapsto x], \qquad
(1,2,3)\ \mapsto\ [x\mapsto y;\quad y\mapsto x^{-1}y^{-1}]$


does *not* give a lifting of $\psi$.



Embedding $S_3$ into $Aut(F_2)$
As Tom Goodwillie noted in his comment, $GL(2,\Bbb Z)$ can be identified with $Out(F\_2)$, so the question can be rephrased in terms of lifting subgroups of $Out(F\_2)$ to $Aut(F\_2)$. There is a Realization Theorem for finite subgroups of $Aut(F\_n)$ and $Out(F\_n)$ which says that such a subgroup can always be realized as a group of symmetries of some finite connected graph with fundamental group $F\_n$, where the symmetries fix a basepoint in the graph in the case of $Aut(F\_n)$. When $n=2$ there are only two graphs to consider, and the relevant one for $S\_3$ is the join of two points with three points. This has two symmetry groups isomorphic to $S\_3$, but only one of these two groups fixes a basepoint, so this should answer the question.


The Realization Theorem is discussed in Karen Vogtmann's survey paper "Automorphism groups of free groups and outer space", section II.6. The references given there are to papers by M. Culler, B. Zimmermann, and D. G. Khramtsov from 1981 to 1984.


