Powering-injectivity is inherited by extensions where the normal subgroup is contained in the hypercenter

Statement
Suppose we have the following:


 * A group $$G$$.
 * A normal subgroup $$H$$ of $$G$$.
 * A set $$\pi$$ of prime numbers.

such that the following are satisfied:


 * $$H$$ is a normal subgroup contained in the hypercenter in $$G$$ (note that this condition is automatically satisfied if $$G$$ is a nilpotent group).
 * Both $$H$$ and $$G/H$$ are $$\pi$$-powering-injective groups. Note that for $$H$$, this is equivalent to being $$\pi$$-torsion-free.

Then, $$G$$ is also a $$\pi$$-powering-injective group.

Facts used

 * 1) uses::Powering-injectivity is inherited by central extensions
 * 2) Something to show that each of the groups $$(H \cap Z^{i+1}(G))/(H \cap Z^i(G))$$ is  $$\pi$$-torsion-free, similar to equivalence of definitions of nilpotent group that is torsion-free for a set of primes.