LCS-powering-invariant subgroup

Definition
A subgroup] $$H$$ of a group $$G$$ is termed LCS-powering-invariant if for every positive integer $$k$$, the $$k^{th}$$ member $$\gamma_k(H)$$ of the lower central series of $$H$$ is a powering-invariant subgroup inside the $$k^{th}$$ member $$\gamma_k(G)$$ of the lower central series of $$G$$.

Facts

 * Powering-invariance does not satisfy lower central series condition