The Sylow Theorems · Superchallenge problem

Suppose a simple group $G$ has order $48=2^4\cdot3$. Since $G$ is simple, its Sylow $2$-subgroup is not normal, forcing $n_2$ to the only other Sylow-III candidate. $G$ then acts by conjugation on its $n_2$ Sylow $2$-subgroups, giving a homomorphism $\varphi:G\to S_{n_2}$ whose image must be transitive (so its order is a multiple of $n_2$) and must divide $|S_{n_2}|$. What is the SMALLEST possible order of $\ker\varphi$ consistent with these constraints?