Let $G$ be a finite solvable group and $F(G)$ its Fitting subgroup. If $F(G)$ is a $p$-subgroup, is $G$ always a split extension over $F(G)$?



Is $G$ always a split extension over Fitting subgroup under certain hypothesis?
This has been answered in the comments, but here is a summary.


There are counterexamples. One such (suggested by Geoff Robinson) is $\texttt{SmallGroup}(48,28)$, with $F(G) \cong Q\_8$. A similar example is $\texttt{SmallGroup}(48,30)$, but here $F(G)$ is elementary abelian of order $8$.


You asked for examples with $|F(G)|$ odd. One such is $\texttt{SmallGroup}(648,531)$, with $F(G$ extraspecial of order $27$. A similar example is $\texttt{SmallGroup}(648,531)$, but here $F(G)$ is elementary abelian of order $27$.


In both of those last two examples, $G/F(G) \cong {\rm SL}(2,3)$, and a corresponding example with $p=5$ would not be solvable. But fortunately you have not requested an example with $p>3$.


