Endo-invariance implies commutator-closed

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 $$H \le G$$ 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.