Finite index implies completely divisibility-closed

Statement
Suppose $$G$$ is a group and $$H$$ is a subgroup of finite index in $$G$$. Then, $$H$$ is a completely divisibility-closed subgroup of $$G$$.

Facts used

 * 1) uses::Poincare's theorem
 * 2) uses::Subgroup of finite group implies completely divisibility-closed
 * 3) uses::Divisibility-closedness satisfies inverse image condition