This is new variant of For $h \in G$ and $\phi \in Aut(G)$ is $\phi^n(h)$ periodic in any finite quotient? thanks to Alon's comment. Since that is the case I'm interested really in, I figured it's worth posting the question again with the condition that $G$ be finitely generated:
Let $G$ be a finitely generated infinite group, and $\phi$ an automorphism of it. Let $N$ be a normal subgroup of $G$ such that $G/N$ is finite. Is it true that for any $h$ in $G$, $\phi^n(h)N$ (as a sequence of elements in $G/N$ for $n=1,2,3,...$) is periodic?