Abstract:
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 $E_6$.