= Solution
Let $A\in GL_2(\mathbb Z)$. If $r-r'\in\mathbb Z^2$, then $A(r-r')\in\mathbb Z^2$, so
$$
f_A(p(r))=p(Ar)
$$
defines a continuous map on $X=\mathbb R^2/\mathbb Z^2$. Since $\det A=\pm1$, the <integral general linear group> contains $A^{-1}$ as another integer matrix, and $f_{A^{-1}}$ is the inverse of $f_A$. Hence $f_A$ is a <homeomorphism>, and $f_A(x_0)=x_0$.
The map $A:\mathbb R^2\to\mathbb R^2$ is the lift of $f_A$ fixing the origin. A loop whose lift ends at $m\in\mathbb Z^2$ is sent to a loop whose lift ends at $Am$. Under the isomorphism in part (a), the induced map is therefore
$$
\boxed{f_{A*}(m)=Am}.
$$
This is precisely the <integral linear automorphism of the torus> associated with $A$.
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