On periodic groups and Shunkov groups that are saturated by dihedral groups and $A_5$

A group is said to be periodic, if any of its elements is of finite order. A Shunkov group is a group in which any pair of conjugate elements generates Finite subgroup with preservation of this property when passing to factor groups by finite Subgroups. The group $G$ is saturated with groups from the set of groups $X$ if any A finite subgroup $K$ of $G$ is contained in the subgroup of $G$, Isomorphic to some group in $X$. The paper establishes the structure of periodic groups And Shunkov groups saturated by the set of groups $\mathfrak {M}$ consisting of one finite simple non-Abelian group $A_5$ and dihedral groups with Sylow $2$-subgroup of order $2$. It is proved that A periodic group saturated with groups from $\mathfrak {M},$ is either isomorphic to a prime Group $A_5$, or is isomorphic to a locally dihedral group with Sylow $2$ subgroup of order $2$. Also, the existence of the periodic part of the Shunkov group saturated with groups from the set $\mathfrak {M}$ is proved, and the structure of this periodic part is established.