Periodic groups saturated with finite simple unitary groups of degree 4 over finite fields of odd characteristic

Suppose that $M$ is some nonempty set of finite groups. A group $G$ is saturated with groups from $M$ if each finite subgroup of $G$ lies in a subgroup isomorphic to an element of $M$. We prove that a periodic group with locally finite centralizers of involutions which is saturated with simple unitary groups of degree $4$ over finite fields of fixed odd characteristic $p$ is isomorphic to a simple unitary group of degree $4$ over some locally finite field of characteristic $p$.