 Solution

The map $z\mapsto z-w$ sends $w$ to $0$, proving transitivity (with inversion handling $\infty$). Fixed points satisfy the quadratic $c{z}^2+(d-a)z-b=0$, with multiplicity and $\infty$ included, so a nonidentity has one or two. The stabiliser of $\infty$ is $S=\{z\mapsto az+b:a\ne 0\}$.