D8 is not potentially fully invariant in D16

Statement
Suppose $$G$$ is the dihedral group of order 16:

$$G := \langle a,x \mid a^8 = x^2 = e, xax = a^{-1} \rangle$$

and $$H$$ is a subgroup isomorphic to the dihedral group of order 8:

$$H := \langle a^2, x \rangle$$

Then, $$H$$ is not a potentially fully invariant subgroup of $$G$$.

Facts used

 * 1) uses::Fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant