Hall subgroups of odd order in finite groups

We complete the description of Hall subgroups of odd order in finite simple groups initiated by F.Gross, and as a consequence, bring to a close the study of odd order Hall subgroups in all finite groups modulo classification of finite simple groups. In addition, it is proved that for every set $\pi$ of primes, an extension of an arbitrary $D_\pi$-group by a $D_\pi$-group is again a $D_\pi$-group. This result gives a partial answer to Question 3.62 posed by L. A. Shemetkov in the “Kourovka Notebook”.