On some actions of the symmetric group $\mathbb S_4$ on $K3$ surfaces

In the talk, I'll consider the actions of the symmetric group $\mathbb S_4$ on $K3$ surfaces $X$ having the following property:

$(*)$ there exists an equivariant bi-rational contraction $\overline c: X\to \overline X$ to a $K3$ surface $\overline X$ with $ADE$-singularities such that $\overline X/\mathbb S_4\simeq \mathbb P^2$.

I'll show that up to equivariant deformations there exist exactly 15 such actions and these actions can be realized as the actions of the Galois group on the Galois normal closures of the dualizing coverings of the projective plane associated with rational quartics having no singularities of types $A_4$, $A_6$ and $E_6$.