Is there some sort of classification of invertible finite groups?

Question

Let’s call a group $G$ invertible, if $\forall H \triangleleft G$, there $\exists K \triangleleft G$, such, that $K \cong \frac{G}{H}$ and $H \cong \frac{G}{K}$

All finite abelian groups are known to be invertible. All simple groups also are invertible.

However the vast majority of finite groups seems to be non-invertible: that includes symmetric groups, dihedral groups, holomorphs of cyclic groups, generalised quaternion groups and so on…

So my question is:

Is there some sort of classification of invertible finite groups?