Characteristic not implies powering-invariant in nilpotent group

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup of nilpotent group) need not satisfy the second subgroup property (i.e., powering-invariant subgroup of nilpotent group)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about characteristic subgroup of nilpotent group|Get more facts about powering-invariant subgroup of nilpotent group

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup of nilpotent group but not powering-invariant subgroup of nilpotent group|View examples of subgroups satisfying property characteristic subgroup of nilpotent group and powering-invariant subgroup of nilpotent group

Statement

It is possible to have a nilpotent group $G$ and a characteristic subgroup $H$ of $G$ such that $H$ is not a powering-invariant subgroup of $G$ . In other words, there exists a prime number $p$ such that every element of $G$ has a unique $p^{th}$ root, but there are elements of $H$ whose $p^{th}$ roots are outside $H$ .