Let $G$ be a finite group and let $C$ be a cyclic subgroup such that $N_G(C) = C$. Such a subgroup is in particular a Carter subgroup and Vdovin (*) has shown that any two such subgroups are conjugate in an arbitrary finite group. Moreover, Zhang (**) showed that a group with cyclic self-normalizing subgroup cannot be non-abelian simple.
I could not come up with a counterexample to the following question so far:
Is a finite group with self-normalizing cyclic subgroup already solvable?
In the paper by Zhang he mentions that the theorem he proves had been a long-standing conjecture. However, he does not give any reference. Can you point me to some previous work on this conjecture or who first formulated it?
(*) Vdovin, E.P., Carter subgroups of finite almost simple groups. Algebra Logika 46, No. 2, 157-216 (2007), arXiv:math/0602156
(**) Zhang, Guangxiang, On self-normalizing cyclic subgroups. J. Algebra 127, No.2, 255-258 (1989).