Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has finitely many orbits, and every point stabiliser $G_x$ has finite orbits. Now consider a permutation $\tau\in\operatorname{Sym}(X)$ of finite order, and let $H=\langle G,\tau\rangle$. Is it necessarily true that every point stabiliser $H_x$ has finite orbits?

The situation I'm particularly interested in is when $\tau$ is a cycle with one element in each orbit of $G$, such that $H$ is transitive.