On exceptional pq-groups

A finite group G is called exceptional if for a Galois extension F/k of number fields with the Galois group G, in the Brauer-Kuroda relation of the Dedekind zeta functions of fields between k and F, the zeta function of F does not appear. In the present paper we describe effectively all exceptional groups of orders divisible by exactly two prime numbers p and q, which have unique subgroups of orders p and q.