I want an example of a group $G$, a normal subgroup $H$, and a prime number $p$, such that:

- $G$ is powered over $p$, i.e., every element of $G$ has a unique $p^{th}$ root in $G$.
- $H$ is also powered over $p$, i.e., every element of $H$ has a unique $p^{th}$ root in $H$.
- The quotient group $G/H$ is not powered over $p$. Since the above conditions already guarantee the existence of $p^{th}$ roots, what I want should fail is the uniqueness condition.

While I suspect that an example exists, the example seems hard to construct, because of the following constraints I worked out for any example:

- $H$ must be infinite and have infinite index in $G$ (i.e., neither $H$ nor $G/H$ can be finite).
- $H$ cannot be contained in the hypercenter of $G$ (the hypercenter is the subgroup at which the upper central series stabilizes). This rules out any example involving $G$ abelian or nilpotent.
- $H$ cannot have a complement (i.e., be part of a semidirect product) in $G$.

The proofs of all these assertions are straightforward, but I'll be happy to provide proofs if they are unclear to readers.

If you find a proof that no such example exists, that would be great to have too.