Subgroup whose derived subgroup equals its intersection with whole derived subgroup

Definition
A subgroup $$H \le G$$ is termed a subgroup whose derived subgroup equals its intersection with whole derived subgroup or subgroup whose commutator subgroup equals its intersection with whole commutator subgroup if $$[H,H] = H \cap [G,G]$$.

Stronger properties

 * Weaker than::Conjugacy-closed Hall subgroup:
 * Weaker than::Perfect subgroup
 * Weaker than::Direct factor
 * Weaker than::Retract:
 * Weaker than::Cocentral subgroup:

Weaker properties

 * Stronger than::Subgroup whose focal subgroup equals its intersection with the commutator subgroup
 * Stronger than::Subgroup whose focal subgroup equals its commutator subgroup

Metaproperties
If $$H \le K \le G$$ are groups such that $$[K,K] = K \cap [G,G]$$ and $$[H,H] = H \cap [K,K]$$, then $$[H,H] = H \cap [G,G]$$.

If $$H \le K \le G$$ such that $$[H,H] = H \cap [G,G]$$, then we also have $$[H,H] = H \cap [K,K]$$.