Endo-invariance property

Symbol-free definition
A subgroup property $$p$$ is termed an endo-invariance property if for any group, there is a collection of endomorphisms of the group such that a subgroup has property $$p$$ in the group if and only if it is invariant under all the endomorphisms in that collection.

In terms of the function restriction formalism
A subgroup property $$p$$ is termed an endo-invariance property if it has a function restriction expression of the form:

$$a \to$$ Function

where $$a$$ is a property of endomorphisms (in other words, $$a$$ is a function property satisfied only by endomorphisms).

Equivalently, it can be expressed in the form:

$$a \to$$ Endomorphism

Equivalence of definitions
The equivalence of the various definitions follows from this observation: restriction of endomorphism to invariant subgroup is endomorphism.