  Given a construction $f$ on groups, we say that a group $G$ is
\textit{$f$-realisable} if there is a group $H$ such that $G\cong f(H)$, and
\textit{completely $f$-realisable} if there is a group $H$ such that $G\cong
f(H)$ and every subgroup of $G$ is isomorphic to $f(H_1)$ for some subgroup
$H_1$ of $H$ and vice versa.
  Denote by $L(G)$ the absolute center of a group $G$, that is the set of
elements of $G$ fixed by all automorphisms of $G$. By using the structure of
the automorphism group of a ZM-group, in this paper we prove that cyclic groups
$C_N$, $N\in\mathbb{N}^*$, are completely $L$-realisable.
