Extended Mislin genus

Definition
Let $$G$$ be a nilpotent group. The extended Mislin genus of $$G$$ is defined as the collection of all nilpotent groups $$H$$ with the property that for every prime number $$p$$, the $$p$$-localizations $$G$$ and $$H$$ are isomorphic groups. Here, by $$p$$-localization, we mean the $$p'$$-powering, i.e., we introduce powering by all primes other than $$p$$, i.e., we consider the image of the initial homomorphism to pi-powering groups.