Why is this a group action?

Question

Let $G$ be a group and let $H$ be an infinite cyclic normal subgroup of $G$ of finite index. Let $K$ be the centralizer of $H$ in G, $$K=C_G(H)$$ and suppose that the index of $K$ in $G$ is 2. Let $E$ be a copy of the real line, with $H$ acting on $E$ by translations, so that a generator $h$ of $H$ acts as translation by 1. If $k\in K$, define the action of $k$ as translation by $m/n$, where $n$ is the least integer such that $k^n=h^m$ for some $m$. Why does this defines an action of $K$ on $E$ extending the action of $H$ on $E$?