 Solution

Choose a Möbius map $T$ with $T(z_1)=0$ and $T(z_2)=\infty$, for example by normalizing the ordered triple $(z_1,z_3,z_2)$. Conjugation $h\mapsto T\circ h\circ T^{-1}$ takes the requested subgroup to the subgroup fixing zero and infinity individually.

Such a transformation must be $z\mapsto cz$ with $c\in {\mathbb{C}}^{\ast }$: the zero condition removes the constant numerator term and the infinity condition removes the nonconstant denominator term. Conversely every such dilation is Möbius and fixes the two points. Composition multiplies its nonzero parameter.This identifies the Möbius pointwise stabilizer of two points. Allowing an interchange of the two fixed points would give a larger group, but those interchanges are excluded by the stated equalities.