  In \cite[Serra, Vena, Extremal families for the Kruskal-Katona
theorem]{sv21}, the authors have shown a characterization of the extremal
families for the Kruskal-Katona Theorem. We further develop some of the
arguments given in \cite{sv21} and give additional properties of these extremal
families. F\"uredi-Griggs/M\"ors theorem from 1986/85 \cite{furgri86,mors85}
claims that, for some cardinalities, the initial segment of the
colexicographical is the unique extremal family; we extend their result as
follows: the number of (non-isomorphic) extremal families strictly grows with
the gap between the last two coefficients of the $k$-binomial decomposition.
  We also show that every family is an induced subfamily of an extremal family,
and that, somewhat going in the opposite direction, every extremal family is
close to being the inital segment of the colex order; namely, if the family is
extremal, then after performing $t$ lower shadows, with $t=O(\log(\log n))$, we
obtain the initial segment of the colexicographical order. We also give a
``fast'' algorithm to determine whether, for a given $t$ and $m$, there exists
an extremal family of size $m$ for which its $t$-th lower shadow is not yet the
initial segment in the colexicographical order.
  As a byproduct of these arguments, we give yet another characterization of
the families of $k$-sets satisfying equality in the Kruskal--Katona theorem.
Such characterization is, at first glance, less appealing than the one in
\cite{sv21}, since the additional information that it provides is indirect.
However, the arguments used to prove such characterization provide additional
insight on the structure of the extremal families themselves.
