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

  • $G$ is $p$-powered, i.e., every element of $G$ has a unique $p^{th}$ root in $G$.
  • $H$ is not $p$-powered. In this case, it would mean that there exists an element of $H$ that does not have any $p^{th}$ root in $H$.

Note that there do not exist any abelian examples, because in an abelian $p$-powered group, the $p^{th}$ root map is an automorphism and hence preserves every characteristic subgroup.

The intuitive reasoning for why there shouldn't be any examples: even though the $p^{th}$ root map is not an automorphism, it seems possible generally to find automorphisms that behave a lot like this map to a sufficient extent that characteristic subgroups must be preserved under taking $p^{th}$ roots.

The intuitive reasoning for why there may well be examples: nilpotent groups can in principle rigidify a lot of powering structure.

I don't know which side to believe.

PS: There are certainly characteristic subgroups of non-nilpotent groups with this property. For instance, let $G = GA^+(1,\mathbb{R})$ (the identity component of the affine group of degree one over the reals, i.e., maps of the form $x \mapsto ax + b, a > 0, a,b \in \mathbb{R}$), and let $H$ be the subgroup comprising those elements where $a$ is rational. $H$ is characteristic in $G$, $G$ is powered over all primes, and $H$ is not powered over any prime.

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