LCS-characteristic subgroup

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