Suppose that we have a finitely generated residually finite group $G = \langle g_1,\ldots,g_r \rangle$ with polynomial growth. Let $H$ be a subgroup of $G$ with finite index $m$. Let $\phi$ be an automorphism on $G$.

**Question**: What is the bound for the *smallest $n \in \mathbb{N} \setminus \{0\}$, such that $\phi ^n(H) = H$*?

**My thought so far**: We know that $G$ has at most $(m!)^r$ number of subgroups with index $m$, so $n \leq (m!)^r$. Are there any known results we can use to improve this bound? Is it possible to show that $n$ is bounded by a polynomial (or even linear function) in $m$, i.e. $ n = \mathcal{O}_r(m^d)$ for some $d$?

If $G$ doesn't have polynomial growth, then $n$ might not be bounded by a polynomial function in $m$ because of this example.