By a Frobenius system on a finite group $G$, we mean the data,
for each maximal solvable subgroup $M$ of $G$, of a normal
subgroup $\mathcal{F}(M)$ of $M$, satisfying some of the properties
of a Frobenius kernel, and subject to certain additional conditions.
We prove that a finite group with a Frobenius system is either
solvable (in which case we get a complete description), or
isomorphic to $\mathit{SL}_{2}(K)$ (for $K$ a finite field
of characteristic 2) or to a Suzuki group. The respective
possibilities for the mapping $\mathcal{F}$ are then determined.
This extends a previous result of ours
(Nagoya Math. J. 165 (2002), 117--121) by removing the condition that each $M/\mathcal{F}(M)$
be abelian. Curiously enough, the Feit-Thompson Theorem is
used in the proof.