Prove that two groups act in the same way

Question

Let $G$ and $H$ are finite cyclic groups acting on a set $A$. The properties (number of orbits and their length) of $G:A$ are known. About $H$ I know only that it is finite and cyclic (don't know the order, but it is supposingly the same as order of $G$) and how it acts on each element of $A$. I want to prove that $H:A$ has the same number of orbits and lengths as $G:A$, but the only approach I see is to construct a bijective mapping $\varphi:A\to A$ such that $ga = \varphi^{-1}h \varphi a$, where $a\in A$, $g\in G$, $h\in H$. Are there other general ideas to solve this problem?