  It was shown by Mukai that the maximum order of a finite group acting
faithfully and symplectically on a K3 surface is $960$ and that the group is
isomorphic to the group $M\_{20}$. Then Kondo showed that the maximum order of
a finite group acting faithfully on a K3 surface is $3\,840$ and this group
contains the Mathieu group $M\_{20}$ with index four. Kondo also showed that
there is a unique K3 surface on which this group acts faithfully, which is the
Kummer surface $\Km(E\_i\times E\_i)$. In this paper we describe two more K3
surfaces admitting a big finite automorphism group of order $1\,920$, both
groups contains $M\_{20}$ as a subgroup of index 2. We show moreover that these
two groups and the two K3 surfaces are unique. This result was shown
independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with
the aim of classifying all the finite groups acting faithfully on K3 surfaces
with maximal symplectic part.
