Another group [can be found][1] in Alan Logan's paper (PAMS 2016): it is a 


  [1]: https://www.dropbox.com/s/xhphrtkya9rjcp1/Screenshot%202017-03-19%2019.21.54.png?dl=0

The example is the two generator, one relator group

$$ \langle G = a, b; (a^{−2}ba^44ba^{−3}ba^5b)^n\rangle.$$

**EDIT** This has nothing to do with the above, but I don't want to add a third answer :) My discussion with Henry in the comments to the OP, might suggest that $Out(PGL(2, \mathbb{Z})$ might be trivial. Apparently, this was proved by Hua and Reiner in 1951-1952, and then disproved(!) by Joan Dyer in 1978, where she constructed an outer automorphism (known ever since as the Dyer automorphism, so $Out(PGL(2, \mathbb{Z}) = C_2;$ the "so" is not trivial but true.

For completeness, her automorphism $\mathcal{D}$ is described as follows. The generators of $PGL(2, \mathbb{Z})$ are:

$$S = \pm \begin{pmatrix}0 &1\\ -1 & 0\end{pmatrix}, T=\pm \begin{pmatrix}1 &1\\ 0 & 1\end{pmatrix},B=\pm \begin{pmatrix}0 &1\\ 1 & 0\end{pmatrix}.$$
Then the automorphism sends $S, T, B$ to $SB, TB, B.$