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