A group $G$ is said to be saturated with groups in a set $X$ provided that every finite subgroup $K\leqslant G$ can be embedded in $G$ into a subgroup $L$ isomorphic to a group in $X$. It is shown that a torsion group with a finite dihedral Sylow 2-subgroup which is saturated with finite simple nonabelian groups is locally finite and isomorphic to $L_2(P)$ (Theorem 1.1). It is proven that a torsion group saturated with finite Ree groups is locally finite and isomorphic to a Ree group (Theorem 1.2).