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

The example is the two generator, one relator group

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

with $n>1$.

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 $SB, TB, B.$

A countably-infinite family of groups is given at the end of Section 6 of a paper of A. D. Logan¹: for each natural number $n>1$, the two-generator-, one-relator group

$$G_n := \langle a, b; (a^{−2}ba^4ba^{−3}ba^5b)^n\rangle$$

has trivial outer automorphism group.

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 $SB, TB, B.$

¹_{A. D. Logan: The outer automorphism groups of two-generator, one-relator groups with torsion. Proc. Amer. Math. Soc. 144 (2016) 4135-4150}

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

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 $SB, TB, B.$

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

The example is the two generator, one relator group

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

with $n>1$.

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 $SB, TB, B.$

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

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 $SB, TB, B.$

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

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 $SB, TB, B.$