Endo-invariance implies commutator-closed

This article gives the statement and possibly, proof, of an implication relation between two subgroup metaproperties. That is, it states that every subgroup satisfying the first subgroup metaproperty (i.e., Endo-invariance property (?)) must also satisfy the second subgroup metaproperty (i.e., Commutator-closed subgroup property (?))
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Statement

Let $\alpha$ be an endomorphism property: a property that evaluates to true or false given any group and endomorphism of that group. Suppose $p$ is the endo-invariance property arising from $\alpha$; in other words:

$p=\alpha \to$ Function

is the property of being a subgroup $H\leq G$ such that every endomorphism of $G$ satisfying property $\alpha$ restricts to an endomorphism of $H$. Then, $p$ is a commutator-closed subgroup property: the commutator of two subgroups, each satisfying $p$ in the whole group, must also satisfy $p$ in the whole group.