Locally finite groups with prescribed structure of finite subgroups

Let $\mathfrak{M}$ be a set of finite groups. Given a group $G$, denote the set of all subgroups of $G$ isomorphic to the elements of $\mathfrak{M}$ by $\mathfrak{M}(G)$. A group $G$ is called saturated by groups in $\mathfrak{M}$ or by $\mathfrak{M}$ for brevity, if each finite subgroup of $G$ lies in some element of $\mathfrak{M}(G)$. We prove that every locally finite group $G$ saturated by $\mathfrak{M}=\{GL_m(p^n)\}$, with $m > 1$ fixed, is isomorphic to $GL_m(F)$ for a suitable locally finite field $F$.