Rational curves on K3 surfaces of small genus

Let ${\mathfrak{B}}_g$ denote the moduli space of primitively polarized $K3$ surfaces $(S,H)$ of genus $g$ over $\mathbb{C}$. It is well-known that ${\mathfrak{B}}_g$ is irreducible and that there are only finitely many rational curves in $\mathrm{\mid }H\mathrm{\mid }$ for any primitively polarized $K3$ surface $(S,H)$. So we can ask the question of finding the monodromy group of such curves. The case of $g=2$ essentially follows from the results of Harris \cite{Ha} to be the full symmetric group $S_{324}$, here we solve the case $g=3$ and $4$.