Plane rational quartics and K3 surfaces

We study actions of the symmetric group $\mathbb S_4$ on K3 surfaces $X$ that satisfy the following condition: there exists an equivariant birational contraction $\overline r\colon X\to\overline X$ to a K3 surface $\overline X$ with ADE singularities such that the quotient space $\overline X/\mathbb S_4$ is isomorphic to $\mathbb P^2$. We prove that up to smooth equivariant deformations there exist exactly 15 such actions of the group $\mathbb S_4$ on K3 surfaces, and that these actions are realized as actions of the Galois groups on the Galoisations $\overline X$ of the dualizing coverings of the plane which are associated with plane rational quartics without $A_4$, $A_6$, and $E_6$ singularities as their singular points.