Calculating Clifford classes for characters containing a linear character once

Given a finite group G, a normal subgroup N of G and an irreducible character χ of G, the Clifford class of χ describes the Clifford theory including Schur indices of χ with respect to the normal subgroup N. Assume there is a subgroup U of N and a linear character θ of U with multiplicity 1 in the restriction of χ to U. We describe a method to calculate the Clifford class of χ in certain cases. This method helps, in particular, in the calculation of the Schur indices of the irreducible characters of the groups contained between the general linear and the special linear groups.