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.