${SR}$-groups of order $2^np^m$ with dihedral Sylow 2-subgroup

The structure of ${SR}$-groups with dihedral Sylow $2$-subgroup modulo Frattini subgroup is described. It is proved that if a group $G$ is a non-supersolvable ${SR}$-group of order $2^np^m$ with dihedral Sylow $2$-subgroup, $p$ is Mersenne prime.