Galois lines for a canonical curve of genus 4, III: non-cyclic Galois lines

Let $C\subset {\mathbb{P}}^3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic zero. For a line $l\subset {\mathbb{P}}^3$, we consider the projection ${\pi }_l:C\to {\mathbb{P}}^1$ from $l$ and the induced extension of function fields ${\pi }_l^{\ast }:k({\mathbb{P}}^1)\hookrightarrow k(C)$. A line $l$ is called an \emph{$S_3$-line} (resp. a \emph{$K_4$-line}) if the extension $k(C)\mathrm{/}{\pi }_l^{\ast }(k({\mathbb{P}}^1))$ is Galois and its Galois group is isomorphic to the symmetric group $S_3$ on three letters (resp. the Klein four-group $K_4$). We prove that the number of $S_3$-lines (resp.\ $K_4$-lines) is at most $10$ (resp.\ $15$).