Groups whose non-normal subgroups are either nilpotent or minimal non-nilpotent

Let $\mathfrak{N}\mathfrak{i}\mathfrak{l}$ be the class of nilpotent groups and $G$ be a group. We call $G$ a meta-$\mathfrak{N}\mathfrak{i}\mathfrak{l}$-Hamiltonian group if any of its non-$\mathfrak{N}\mathfrak{i}\mathfrak{l}$ subgroups is normal. Also, we call $G$ a para-$\mathfrak{N}\mathfrak{i}\mathfrak{l}$-Hamiltonian group if $G$ is a non-$\mathfrak{N}\mathfrak{i}\mathfrak{l}$ group and every non-normal subgroup of $G$ is either a $\mathfrak{N}\mathfrak{i}\mathfrak{l}$-group or a minimal non-$\mathfrak{N}\mathfrak{i}\mathfrak{l}$ group. In this paper we investigate the class of finitely generated meta-$\mathfrak{N}\mathfrak{i}\mathfrak{l}$-Hamiltonian and para-$\mathfrak{N}\mathfrak{i}\mathfrak{l}$-Hamiltonian groups.