Epicenter not is fully invariant

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., epicenter) does not always satisfy a particular subgroup property (i.e., fully invariant subgroup)
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Statement

It is possible to have a group $G$ such that the epicenter $Z^{*}(G)$ of $G$ is not a fully invariant subgroup of $G$.

Proof

Further information: nontrivial semidirect product of Z4 and Z4, subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4, subgroup structure of nontrivial semidirect product of Z4 and Z4

Suppose $G$ is the nontrivial semidirect product of Z4 and Z4, given explicitly as follows, where $e$ denotes the identity element:

$G:=\langle x,y\mid x^{4}=y^{4}=e,yxy^{-1}=x^{3}\rangle$

As we can see from the subgroup structure of $G$, the biggest quotient of $G$ that is a capable group is dihedral group:D8, and this arises as the quotient of $G$ by the subgroup $H=\langle y^{2}\rangle$, which is a subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4. Thus, $H$ is the epicenter of $G$.

$H$ is not fully invariant in $G$. For instance, it is not invariant under the endomorphism with kernel $\langle x\rangle$ that sends $y$ to $x$.