Let $H$ be a subgroup of $\mathbb{Z}^2$ with finite index $m$. Let $\phi$ be an automorphism on $\mathbb{Z}^2$. Then $\phi$ corresponds to a matrix in $ \operatorname{GL}(2, \mathbb{Z})$.
Question: What is the bound for the smallest $n \in \mathbb{N} \setminus \{0\}$, such that $\phi ^n(H) = H$? Is it possible to get a linear or even sublinear bound?
Thoughts so far: Let $a(m)$ be the number of subgroups in $G$ with index $m$, then $a(m)$ is bounded by a polynomial but not a linear function in $m$. It follows that $n$ is bounded by a polynomial in $m$. I was wondering if this bound can be improved, it seems unlikely for the automorphism to go through all the subgroups of this index.
Any references for this question would be really appreciated, thanks for reading.