Characteristic not implies isomorph-normal in finite group

Statement with symbols
It is possible to have a group $$G$$ and a subgroup $$H$$ of $$G$$ such that $$H$$ is a characteristic subgroup of $$G$$ but is not isomorph-normal in $$G$$: there exists a subgroup $$K$$ of $$G$$ isomorphic to $$H$$ that is not normal in $$G$$.

Example of the dihedral group
Let $$G$$ be the dihedral group of order eight, given by:

$$G = \langle a,x \mid a^4 = x^2 = 1, xax = a^{-1} \rangle$$.

Let $$H$$ be the center of $$G$$. $$H$$ is a subgroup of order two generated by $$a^2$$.


 * $$H$$ is characteristic.
 * $$H$$ is not isomorph-normal: The subgroup $$\langle x \rangle$$ of $$G$$ is isomorphic to $$H$$, but is not normal in $$G$$, because conjugation by $$a$$ sends it to the subgroup $$\langle a^2x \rangle$$.