Finite direct power-closed characteristic is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., finite direct power-closed characteristic subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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Statement

Statement with symbols

Suppose $G$ is a group and $H$ and $K$ are subgroups with $H\leq K\leq G$. Suppose $H$ is a finite direct power-closed characteristic subgroup of $G$. Since characteristic implies normal, we can talk of the quotient group $G/H$. Suppose, further, that $K/H$ is a finite direct power-closed characteristic subgroup of $G/H$. Then, $K$ is also a finite direct power-closed characteristic subgroup of $G$.

Facts used

1. Characteristicity is quotient-transitive: If $A\leq B\leq C$ with $A$ characteristic in $C$ and $B/A$ characteristic in $C/A$, then $B$ is characteristic in $C$.
2. (no link): The fact that $(K/H)^{n}\cong K^{n}/H^{n}$ and $(G/H)^{n}\cong G^{n}/H^{n}$, and the natural embedding from $(K/H)^{n}$ to $(G/H)^{n}$ coincides via these isomorphisms with the natural embedding from $K^{n}/H^{n}$ to $G^{n}/H^{n}$. where ${}^{n}$ stands for the $n^{th}$ direct power.

Proof

Given: A group $G$, a finite direct power-closed characteristic subgroup $H$ of $G$. A subgroup $K$ of $G$ containing $H$ such that $K/H$ is a finite direct power-closed characteristic subgroup of $G/H$. $n$ is a natural number.

To prove: $K^{n}$ is a characteristic subgroup of $G^{n}$ under the natural embedding.

Proof:

Step no.	Assertion	Definitions used	Facts used	Given data used	Previous steps used	Explanation
1	$(K/H)^{n}$ is characteristic in $(G/H)^{n}$	finite direct power-closed characteristic subgroup	--	$K/H$ is finite direct power-closed characteristic in $G/H$	--	Piece together definition and given data.
2	$K^{n}/H^{n}$ is characteristic in $G^{n}/H^{n}$	--	fact (2)	--	step (1)	Using the natural identification indicated by fact (2), step (2) becomes a reformulation of step (1).
3	$H^{n}$ is characteristic in $G^{n}$	finite direct power-closed characteristic subgroup	--	$H$ is finite direct power-closed characteristic in $G$	--	Piece together definition and given data.
4	$K^{n}$ is characteristic in $G^{n}$	--	fact (1)	--	steps (2) and (3)	Piece together. [SHOW MORE] $H^{n}\leq K^{n}\leq G^{n}$, with $H^{n}$ characteristic in $G^{n}$ (step (3)) and $K^{n}/H^{n}$ characteristic in $G^{n}/H^{n}$. Set $A=H^{n}$, $B=K^{n}$, and $C=G^{n}$ in fact (1). We get $K^{n}$ is characteristic in $G^{n}$.