Left-transitively WNSCDIN not implies characteristic

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.