Let $G$ be an 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?
On the one hand my intuition tells me that it's false, but on the other I can't find any counter-examples.