Direct factor satisfies lower central series condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor) satisfying a subgroup metaproperty (i.e., lower central series condition)
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Statement

Suppose $H$ is a subgroup of a group $G$ that is a direct factor of $G$. Then, for every positive integer $k$, the lower central series member $\gamma _{k}(H)$ is a direct factor of $\gamma _{k}(G)$.