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