Free powered nilpotent group for a set of primes

Definition
Suppose $$S$$ is a set, $$\pi$$ is a set of primes, and $$c$$ is a positive integer. The free $$\pi$$-powered class $$c$$ group on $$S$$ is defined as the free algebra on $$S$$ in the variety of $$\pi$$-powered groups that are nilpotent and have nilpotency class $$\le c$$.

Alternatively, it can be described as the unique minimal pi-powered nilpotent group containing the free nilpotent group on $$S$$ of class $$c$$.