Cyclic characteristic implies hereditarily characteristic

Statement
Suppose $$G$$ is a group and $$K$$ is a cyclic characteristic subgroup of $$G$$, i.e., $$K$$ is a cyclic group and is also a fact about::characteristic subgroup of $$G$$. Then, $$K$$ is also a hereditarily characteristic subgroup of $$G$$, i.e., every subgroup $$H$$ of $$K$$ is characteristic in $$G$$.

Related facts

 * Cyclic normal implies hereditarily normal
 * Hereditarily characteristic not implies cyclic in finite
 * SQ-dual::Cyclic-quotient characteristic implies upward-closed characteristic

Facts used

 * 1) uses::Cyclic implies every subgroup is characteristic
 * 2) uses::Characteristicity is transitive: If $$A \le B \le C$$ with $$A$$ characteristic in $$B$$ and $$B$$ characteristic in $$C$$, then $$A$$ is characteristic in $$C$$.

Proof
Given: A group $$G$$ with a cyclic characteristic subgroup $$K$$. A subgroup $$H$$ of $$K$$.

To prove: $$H$$ is characteristic in $$K$$.

Proof:


 * 1) $$H$$ is characteristic in $$K$$: This follows from fact (1).
 * 2) $$H$$ is characteristic in $$G$$: This follows from the previous step, the given datum that $$K$$ is characteristic in $$G$$, and fact (2).