Finite groups of symplectic automorphisms of $K3$ surfaces were first studied by Nikulin (1980), classified by Mukai (1988) relating to the sporadic Mathieu groups in characteristic 0, and extended by Dolgachev-Keum (2009) to positive characteristic in (mainly) the tame case. Based on a joint work with Professor M. Schuett (Leibniz university of Hannover), we will give a partial extension/refinement of these results. In particular, we will obtain a full statement in the special case in the title.

[<6>]