Reductions of algebraic integers

Let K be a number field, and let G   be a finitely generated subgroup of K×K×. Fix some prime number ℓ  , and consider the set of primes pp of K satisfying the following property: the reduction of G   modulo pp is well-defined and has size coprime to ℓ. We are able to give a closed-form expression for the natural density of this set.