  Finite-above inverse monoids are a common generalization of finite inverse
monoids and Margolis--Meakin expansions of groups. Given a finite-above
$E$-unitary inverse monoid $M$ and a group variety $\mathit{U}$, we find a
condition for $M$ and $\mathit{U}$, involving a construction of descending
chains of graphs, which is equivalent to $M$ having an $F$-inverse cover via
$\mathit{U}$. In the special case where $\mathit{U}=\mathit{Ab}$, the variety
of Abelian groups, we apply this condition to get a simple sufficient condition
for $M$ to have no $F$-inverse cover via $\mathit{Ab}$, formulated by means of
the natural parial order and the least group congruence of $M$.
