I can't understand this concept clearly.
For a class $\mathfrak{H}$ of groups, define $b( \mathfrak{H}):= (G \in \mathfrak{C} \ \setminus \mathfrak{H}: G/N \in \mathfrak{H}$ for all $ 1 \neq N \trianglelefteq G ) $. Obviously, $b(\phi)= b( \mathfrak{C})= \phi$.
We say that a class of groups $\mathfrak{B}$ is a boundary if $\mathfrak{B}=b( \mathfrak{H})$ for some classes of groups $ \mathfrak{H}$.
Well, I remember my teacher said that, for example, $S_{3}$ is in the boundary of nilpotent class since it is non-nilpotent group but every subgroup in $S_{3}$ is nilpotent. But I can't actually relate between what my teacher said and the definition above! If this idea is correct, how? If not, can you help me with correcting my misunderstanding and suggest a reference to use? Thank in advance.
(Note: I am only studying finite groups.)