Zeros of S-characters

The concept of S-characters of finite groups was introduced by Zhmud' as a generalisation of transitive permutation characters. Any non-trivial S-character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. J-P. Serre asked whether this generalises to S-characters. We provide a translation of this question into the language of polyhedral geometry and thereby construct many counterexamples, using the capabilities of the computer algebra system OSCAR.