D8 is not potentially fully invariant in D16

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Dihedral group:D8 (?)) not satisfying a particular subgroup property (namely, Potentially fully invariant subgroup (?)) in a particular group or type of group (namely, Dihedral group: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. Fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant