On groups occurring as absolute centers of finite groups

Given a construction $f$ on groups, we say that a group $G$ is $f\mathit{\text{-realisable}}$ if there is a group $H$ such that $G\cong f(H)$, and $\mathit{\text{completely}}f\mathit{\text{-realisable}}$ 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 $C_N$, $N\in {\mathbb{N}}^{\ast }$, are completely $L$-realisable.