Epicenter not is epabelian

Statement
It is possible to have a group $$G$$ such that the epicenter $$Z^*(G)$$ is not an epabelian group. In other words, $$Z^*(Z^*(G))$$ is a proper subgroup of $$Z^*(G)$$.

Related facts

 * Epicenter of finite abelian group is epabelian, which in turn implies that $$Z^*(Z^*(Z^*(G))) = Z^*(Z^*(G))$$.

Proof
We can construct examples within groups of order 32. Explicitly, either of SmallGroup(32,32) and SmallGroup(32,33) can be used as $$G$$. The epicenter in both cases is direct product of Z4 and Z2, whose epicenter is cyclic group:Z2 by the characterization of epicenter of finite abelian group.