Free powered group for a set of primes

Definition
Suppose $$\pi$$ is a set of primes and $$S$$ is a generating set. The free $$\pi$$-powered group on $$S$$, denoted $$F(S,\pi)$$ with $$S$$ identified as a subset of this group, is the unique (up to isomorphism and with embedding of $$S$$) group satisfying the following:


 * 1) This group is $$\pi$$-powered.
 * 2) It has no proper subgroup that contains $$S$$ and is also $$\pi$$-powered.
 * 3) Any set map from $$S$$ to a group $$G$$ extends to a unique group homomorphism from $$F(S,\pi)$$ to $$G$$.