 Möbius transformations permuting three points

The Möbius transformations preserving a three-point set act faithfully on it: one fixing all three is the identity. Every permutation is realized because a Möbius transformation can send any ordered distinct triple to any other, by composing the two normalizations to $0,1,\infty$. For that normalized set the six maps are $z$, $1-z$, $1/z$, $1/(1-z)$, $z/(z-1)$ and $(z-1)/z$. Thus its setwise stabilizer is the symmetric group $S_3$. Normalizing any other three-point set conjugates its stabilizer to this group.