Around 1980, G. Mason announced the classification of a certain subclass of an important class of finite simple groups known as ``quasithin groups''. The classification of the finite simple groups depends upon a proof that there are no unexpected groups in this subclass. Unfortunately Mason neither completed nor published his work. In the Main Theorem of this two-part book (Volumes 111 and 112 of the AMS Mathematical Surveys and Monographs series) the authors provide a proof of a stronger theorem classifying a larger class of groups, which is independent of Mason's arguments. In particular, this allows the authors to close this last remaining gap in the proof of the classification of all finite simple groups. An important corollary of the Main Theorem provides a bridge to the program of Gorenstein, Lyons, and Solomon (AMS Mathematical Surveys and Monographs, Volume 40) which seeks to give a new, simplified proof of the classification of the finite simple groups. Part II of the work (this volume) contains the proof of the Main Theorem, and the proof of the corollary classifying quasithin groups of even type. Part I (Volume 111) contains results which are used in the proof of the Main Theorem. Some of the results are known and fairly general, but their proofs are scattered throughout the literature; others are more specialized and are proved here for the first time.