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. 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.