w x Schur groups were introduced in 8 as a certain kind of ‘‘smallest’’ groups admitting characters with nontrivial Schur index. Indeed every Ž . finite group G having an irreducible character x with m x / 1 involves such Schur groups. They occur as terminal steps by repeated application of Ž . the Brauer]Witt theorem over local fields of characteristic zero . Ž . There is a connection with the Schur subgroup S K of the Brauer Ž . Ž . group Br K of such a field K. This S K has been studied in detail by w x w x Yamada 12 and, quite recently, by Adem and Milgram 1 . From the character-theoretic point of view it appears to be of interest to give explicit Ž . Ž . generators for S K in terms of Schur groups see Section 6 . In the course w x of this investigation the authors noticed that there must be a gap in 8 Ž . Ž concerning the exceptional dyadic Schur groups yielding nontrivial 2-lo. cal index . It turned out that the definition of the Schur groups of type Ž . QD, q was too restrictive. The correct definition will be given in Section 5 below. We take the opportunity to discuss these ‘‘terminal Q -reduc2 Ž . tions’’ once again, and the class of 2-groups involved Theorem 4 and 5 . The Schur groups admit faithful irreducible characters which are induced from linear characters of cyclic normal subgroups. For the sake of clarity we begin our present work by making some general comments on Schur indices in group extensions. This should be compared with Turull w x 11 . Our approach is perhaps more elementary, and we just demonstrate Ž . w x how to use the results on stable Clifford theory 7 . More details may be w x found in the recent thesis 5 of the first-named author.