Some linear actions of finite groups with q′-orbits

Let G be a finite group acting faithfully on a finite vector space M in such a way that the centralizer of every element of M contains a Sylow q-subgroup of G as a central subgroup (for a fixed prime divisor q of vertical bar G vertical bar with (q, vertical bar M vertical bar = 1). Then G is isomorphic to a subgroup of the semi-linear group on M.