Let $G$ be any finite solvable group with Fitting subgroup
$F(G)$. Which conditions on $F(G)$ makes $G$ to be supersolvable?
(It is well-known that any finite solvable group with cyclic Fitting subgroup
is supersolvable). In particular, if $F(G)$ is abelian which conditions on
$F(G)$ (and $G$) are needed for $G$ being supersolvable?



Finite solvable groups with abelian Fitting subgroup
The relation between supersolvability of a finite group and its Fitting
subgroup is that for a finite group $G$ the following are equivalent:


1. $G$ is supersolvable.
2. $G' \leq {\rm Fit}(G)$ and ${\rm Fit}(G)$ is the product of cyclic
 and weak $S$-quasinormal subgroups of $G$ of prime power orders.


Here a subgroup $H$ of $G$ is called *weak* $S$- *quasinormal* in $G$
if, for every $p \in \pi(G)$, there is at least one Sylow $p$-subgroup
of $G$ that permutes with $H$. -- See Theorem 2 in


Piroska Csörgö, [*On Supersolvability of Finite Groups*](http://journals.cambridge.org/download.php?file=%2FGMJ%2FGMJ43_03%2FS0017089501030129a.pdf&code=584e444c4bf13c327ed3eff3d6c22b2f),
 Glasgow Math. J. 43(2001), 327-333.


