Clifford classes for some overgroups of the special linear groups

In [A. Turull, J. Algebra 235 (2001), 275–314], we calculated the Schur index of each of the irreducible characters of the finite special linear groups. In the present paper, we calculate the Schur index of all the irreducible characters of some overgroups of the special linear groups. The overgroups in question are the special linear groups extended by diagonal automorphisms, and the subgroups of the general linear group that contain the special linear group. To each conjugacy class of irreducible characters of the special linear group in each overgroup is associated a Clifford class. The Clifford class controls all the irreducible characters of the overgroup and intermediate subgroups that are related to the given irreducible by Clifford theory. Knowing only the Clifford class, we can parametrize all the irreducible characters of the intermediate subgroups, and compute, for each parametrized irreducible character, its field of character values, as well as its Schur index over each field. We explicitly compute the Clifford class in each case, and deduce from it the information on the Schur index of all the irreducible characters of the overgroups.