The talk is devoted to the actions of the symmetric group $\mathbb S_4$ on $K3$ surfaces $X$
satisfying the following properties:
\begin{itemize}
\item[$(*)$] the action of the alternating group $\mathbb A_4\subset \mathbb S_4$ on $X$ is
symplectic;
\item[$(**)$] there is an equivariant birational contraction $r: X\to \overline X$ to a surface
$\overline X$ with $ADE$singularities such that
$\overline X/\mathbb S_4\simeq \mathbb P^2$.
\end{itemize}
It will be explained that up to smooth equivariant deformations there are exactly 16 such
actions
of $\mathbb S_4$ on the $K3$ surfaces and all these actions are realised as the actions of the
Galois group
on the Galois closures $\overline X$ of the dualising
coverings of the projective plane associated with plane rational quartics having no the
singularities
of the types $A_4$, $A_6$, and $E_6$.
