Let $A=\{0,1,\infty,a_1,\ldots,a_n\}$ and $B=\{0,1,\infty,b_1,\ldots,b_n\}$ be subsets of the Riemann sphere.
Let $\sigma$ be an automorphism of the Riemann sphere, i.e., a Möbius transformation, such that $\sigma(A) = B$.
What can one say about $\sigma$?
Example. Suppose that $n=0$. Then $\sigma$ is an automorphism sending $\{0,1,\infty\}$ to $\{0,1,\infty\}$. So $\sigma$ is either the identity map or $z\mapsto \frac{1}{z}$.
Example. Suppose that $n=1$. Under the hypothesis, we have that $b_1 = a_1$. (The cross ratio of $A$ and $B$ should be equal.) So there should be four possibilities for $\sigma$; one of them being the identity map.