Let $X$ and $Y$ be two copies of $S^2$, and let $A_5$ act on each of them (as a group of rotations). Call these actions $\theta_X$ and $\theta_Y$.

Moreover, let $g \in A_5$ be a fixed element of order $5$ (without loss of generality, the cycle $(12345)$) and suppose that $\theta_X(g)$ is a rotation by $\frac{2 \pi}{5}$ and $\theta_Y(g)$ is a rotation by $\frac{4 \pi}{5}$. These actions are unique (up to rotations of $X$ and $Y$) and correspond to the two distinct irreducible representations of $A_5$ with dimension $3$. Consequently, the following question is well defined:

Does there exist a bijection $f : X \rightarrow Y$ which 'commutes with' these actions, in the sense that:

$$ \theta_Y(h) \circ f = f \circ \theta_X(h) $$

for all elements $h \in A_5$?

Note that if we replace $S^2$ with the (dense) intersection $S^2 \cap \mathbb{Q}[\sqrt{5}]^3$, then there is such a bijection: the field automorphism which interchanges $\sqrt{5}$ and $-\sqrt{5}$. Unfortunately, this does not extend to a field automorphism of $\mathbb{R}$, as there are no nontrivial field automorphisms of $\mathbb{R}$.