Finite direct power-closed characteristic is quotient-transitive

Statement with symbols
Suppose $$G$$ is a group and $$H$$ and $$K$$ are subgroups with $$H \le K \le G$$. 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) uses::Characteristicity is quotient-transitive: If $$A \le B \le C$$ 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: