Subgroup isomorphic to whole group need not be normal

Statement
It is possible to have a group $$G$$ and a subgroup $$H$$ of $$G$$ such that $$H$$ and $$G$$ are isomorphic groups but $$H$$ is not a fact about::normal subgroup of $$G$$.

Related facts

 * Every group is normal in itself

Example of the infinite dihedral group
Suppose $$G$$ is the infinite dihedral group, given by:

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

Suppose $$H$$ is the subgroup of $$G$$ generated by $$a^4$$ and $$x$$.

Then, the map $$\sigma$$ that sends $$a$$ to $$a^4$$ and $$x$$ to $$x$$ is an isomorphism from $$G$$ to $$H$$. Thus, $$H$$ is isomorphic to $$G$$. However, $$H$$ is not a normal subgroup of $$G$$, because conjugation by $$a$$ sends $$x$$ to $$a^2x$$, which is not in $$H$$.