Let \(H\) be a subgroup of a finite group \(G\), let \(\text{cf}(H)\) denote the space of complex class functions on \(H\), and let \(\text{cf}(H)^G\) denote the space of class functions on \(G\) induced from \(H\). A basis of the space \(\text{cf}(H)^G\) is called “good” if the basis vectors are induced from irreducible characters of \(H\) and every character of \(G\) induced from \(H\) is a nonnegative integer combination of the basis vectors. A good basis does not always exists, but when it does it is unique, and determines a canonical partition of \(\text{Irr}(G)\) analogous to Brauer’s \(p\)-blocks (the latter being obtained by taking for \(H\) a \(p\)-complement of the \(p\)-soluble group \(G\)).
G. Navarro proved [in J. Aust. Math. Soc., Ser. A 61, No. 3, 369-376 (1996; Zbl 0874.20006)] that if \(G\) is soluble and \(H\) is a nilpotent injector of \(G\), then a good basis for \(\text{cf}(H)^G\) exists. The present paper refines that result as follows: if \(H/N\) is a nilpotent injector of some soluble quotient \(G/N\) of \(G\), then a good basis for \(\text{cf}(H)^G\) exists.