Local powering-invariance is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., local powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Suppose $G$ is a group and $H,K$ are subgroups with $H\leq K\leq G$. Suppose further that $H$ is a local powering-invariant subgroup of $K$ and $K$ is a local powering-invariant subgroup of $G$. Then, $H$ is a local powering-invariant subgroup of $G$.

Facts used

1. Balanced implies transitive

Proof

Abstract proof

Local powering-invariance has a function restriction expression as the balanced subgroup property:

Local powering $\to$ Local powering

Here, we can define a "local powering" as a function that takes every element to its unique $n^{th}$ root for some $n$ that could vary with the element. Note that any element that lacks a nontrivial unique root can just be sent to itself, using $n=1$.

Fact (1) now gives us the desired result.

Concrete proof (using element-chasing)

This is pretty straightforward.