Epicenter not is fully invariant

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
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 $$x$$.