Left-transitively WNSCDIN not implies characteristic

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., left-transitively WNSCDIN-subgroup) need not satisfy the second subgroup property (i.e., characteristic subgroup)
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Statement

We can have a group $G$ and a left-transitively WNSCDIN-subgroup $H$ of $G$ such that $H$ is not a characteristic subgroup of $G$.

Related facts

• Left-transitively WNSCDIN not implies normal

Proof

Consider any group $G$ with a non-characteristic subgroup $H$ of order two. Clearly, $H$ is left-transitively WNSCDIN, because for any embedding of $G$ in a bigger group $K$, $H$ is a WNSCDIN-subgroup of $K$.

A concrete example of this would be $G$ as a Klein-four group, and $H$ as one of its subgroups of order two.