Showing that two group actions are not equivalent

Question

Let $G$ be a topological group, and $X,Y$ be two $G$-spaces. $X$ and $Y$ are said to be equivalent if there is a $G$-equivariant homeomorphism $f:X\to Y$. Now consider the unit circle $S^1\subset \Bbb C$, and let $\Bbb Z_5\subset S^1$ be the group of 5-th roots of unity. Consider two $\Bbb Z_5$-actions on $S^1$ given by $\gamma\cdot z=\gamma z$ and $\gamma\cdot z=\gamma^2z$ where $\gamma=e^{2\pi i /5}$. In p.35 of Bredon's book Introduction to Compact Transformation Groups, it is written that these two actions are not equivalent, but I can't see why. How can we show that there is no $\Bbb Z_5$-equivariant homeomorphism $f:S^1\to S^1$?