Every group admits an initial homomorphism to a pi-powered group

Statement
Suppose $$G$$ is a group and $$\pi$$ is a set of primes. There exists a $$\pi$$-powered group $$K$$ and a homomorphism of groups $$\alpha:G \to K$$ such that for any $$\pi$$-powered group $$L$$ and any homomorphism $$\varphi:G \to L$$, there is a unique homomorphism $$\theta:K \to L$$ such that $$\varphi = \theta \circ \alpha$$.

We can think of the functor that sends $$G$$ to the group $$K$$ as the free $$\pi$$-powering functor. It is the left-adjoint functor to the forgetful functor from $$\pi$$-powered groups to groups.

In the localization terminology
The term $$\pi$$-local is sometimes used to refer to a group that is powered over all the primes not in $$\pi$$. The functor described above is, in that context, termed the $$\pi'$$-localization functor. By $$\pi'$$ we mean the complement of $$\pi$$ in the set of all primes.

Related facts

 * The free powered group for a set of primes can be thought of as being obtained from the abstract free group by applying the free $$\pi$$-powering functor.

Embeddability results
We are interested in cases where the canonical homomorphism from the group to its $$\pi$$-powering is injective, and in related questions, some of which are explored below.