Left-transitively WNSCDIN not implies normal

Statement
We can have a group $$G$$ and a left-transitively WNSCDIN-subgroup $$H$$ of $$G$$ that is not normal in $$G$$.

Example of a subgroup of order two
Let $$G$$ be any group with a non-normal subgroup $$H$$ of order two. Then, $$H$$ is left-transitively WNSCDIN in $$G$$.

For a concrete example, take $$G$$ to be a dihedral group and $$H$$ to be a subgroup of order two generated by a reflection element.