= Solution
For the punctured torus, $G=\langle\alpha,\beta\rangle\cong F_2$ and $G^{\mathrm{ab}}\cong\mathbb Z^2$. Dehn twists about the two standard curves act on homology, in suitable oriented bases, by
$$
\begin{pmatrix}1&0\\1&1\end{pmatrix},
\qquad
\begin{pmatrix}1&-1\\0&1\end{pmatrix}.
$$
The supplied result says that these matrices generate $SL_2(\mathbb Z)$. Thus the image of
$$
q\circ p:\operatorname{Mod}(T_*^2)\longrightarrow GL_2(\mathbb Z)
$$
contains the infinite group $SL_2(\mathbb Z)$. Its intermediate group $\operatorname{Out}(F_2)$ must therefore be infinite.
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