K3 surfaces with maximal finite automorphism groups

In the $80$'s Nikulin classified all the finite abelian groups acting symplectically on a K3 surface and his results inspired an intensive study of automorphism groups of K3 surfaces. It was shown by Mukai that the maximum order of a finite group acting symplectically on a K3 surface is $960$ and that the group is isomorphic to the Mathieu group $M_{20}$. Then Kondo showed that the maximum order of a finite group acting on a K3 surface is $3840$ and this group contains the Mathieu group with index four. Kondo showed also that there is a unique K3 surface on which this group acts, which is a Kummer surface. I will present recent results on finite groups acting on K3 surfaces, that contain strictly the Mathieu group and I will classify them. I will show that there are exactly three groups and three K3 surfaces with this property. This is a joint work with C. Bonnafé.