Base of a wreath product implies right-transitively conjugate-permutable

Property-theoretic statement
The following equivalent statements are true:


 * 1) The subgroup property of being the base of a wreath product implies, i.e., is stronger than, the subgroup property of being a right-transitively conjugate-permutable subgroup.
 * 2) Applying the composition operator to the property of being a conjugate-permutable subgroup and the property of being the base of a wreath product, yields a property stronger than the property of being a conjugate-permutable subgroup.

Conjugate-permutable $$*$$ Base of a wreath product $$\le$$ Conjugate-permutable

Verbal statement
The following equivalent statements are true:


 * 1) Any base of a wreath product in a group is a right-transitively conjugate-permutable subgroup.
 * 2) A conjugate-permutable subgroup of the base of a wreath product is conjugate-permutable in the whole group.

Statement with symbols
The following equivalent statements are true:


 * 1) If $$K$$ is the base of a wreath product in a group $$G$$, then $$K$$ is a right-transitively conjugate-permutable subgroup of $$G$$.
 * 2) If $$H$$ is a conjugate-permutable subgroup of $$K$$ and $$K$$ is the base of a wreath product in $$G$$, then $$H$$ is conjugate-permutable in $$G$$.

Related facts

 * Base of a wreath product implies right-transitively 2-subnormal
 * Base of a wreath product is transitive