Characterization of simple symplectic groups of degree $4$ over locally finite fields of characteristic $2$ in the class of periodic groups

Suppose that each finite subgroup of even order of a periodic group containing an element of order $2$ lies in a subgroup isomorphic to a simple symplectic group of degree $4$ over some finite field of characteristic $2$. We prove that in that case the group is isomorphic to a simple symplectic group $S_4(Q)$ over some locally finite field $Q$ of characteristic $2$.