Periodic groups saturated with the linear groups of degree $2$ and the unitary groups of degree $3$ over finite fields of odd characteristic

Let $\mathfrak{M}$ denote the set of the simple $3$-dimensional unitary groups $U_3$ and the simple linear groups $L_2$ over finite fields of odd characteristic. We prove that each periodic group saturated with groups in $\mathfrak{M}$ is locally finite and isomorphic to either $U_3(Q)$ or $L_2(Q)$ for a suitable locally finite field $Q$ of odd characteristic.