Shunkov groups saturated with almost simple groups

A group $G$ is called a Shunkov group (a conjugate biprimitive finite group) if, for any of its finite subgroups $H$ in the factor group $N_G(H)/H$, every two conjugate elements of prime order generate a finite subgroup. We say that a group is saturated with groups from the set $\mathfrak{M}$ if any finite subgroup of the given group is contained in its subgroup isomorphic to some group in $\mathfrak{M}$. We show that a Shunkov group $G$ which is saturated with groups from the set $\mathfrak{M}$ possessing specific properties, and contains an involution $z$ with the property that the centralizer $C_G(z)$ has only finitely many elements of finite order will have a periodic part isomorphic to one of the groups in $\mathfrak{M}$. In particular, a Shunkov group $G$ that is saturated with finite almost simple groups and contains an involution $z$ with the property that the centralizer $C_G(z)$ has only finitely many elements of finite order will have a periodic part isomorphic to a finite almost simple group.