Completely realisable groups

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. In this paper, we determine completely \({\rm Aut}\)-realisable groups. We also study \(f\)-realisable groups for \(f=Z,F,M,D,\Phi\), where \(Z(H)\), \(F(H)\), \(M(H)\), \(D(H)\) and \(\Phi(H)\) denote the center, the Fitting subgroup, the Chermak-Delgado subgroup, the derived subgroup and the Frattini subgroup of the group \(H\), respectively.