Function restriction formalism chart

Subgroup property	Function restriction expression	Endo-invariance	Balanced	Invariance	Left-inner	Quotient-hereditary invariance
Normal subgroup	Inner $\to$ Aut, Inner $\to$ Function	Yes	No	Yes	Yes	Yes
Characteristic subgroup	Aut $\to$ Aut, Aut $\to$ Function	Yes	Yes	Yes	No	No
Strictly characteristic subgroup	Surj. End $\to$ Function, Surj. End. $\to$ End.	Yes	No	Yes	No	Yes
Fully invariant subgroup	End. $\to$ End.	Yes	Yes	Yes	No	Yes
Injective endomorphism-invariant subgroup	Inj. End.$\to$ End., Inj. end. $\to$ Inj. End.	Yes	Yes	Yes	No	No
Central factor	Inner $\to$ Inner	No	Yes	No	Yes	No
Transitively normal subgroup	Normal $\to$ Normal, Inner $\to$ Normal	No	Yes	No	Yes	No
Conjugacy-closed normal subgroup	Class $\to$ Class, Inner $\to$ Class	No	Yes	No	Yes	?
Retraction-invariant subgroup	Retraction $\to$ Retraction, Retraction $\to$ Function	Yes	Yes	Yes	No	Yes
Powering-invariant subgroup	Rational power map $\to$ Rational power map, Rational power map $\to$ Function	No	Yes	Yes	No	?
Local powering-invariant subgroup	Local powering $\to$ Local powering, Local powering $\to$ Function	No	Yes	Yes	No	?

Note:

• Endo-invariance implies join-closed. For full proof, refer: Endo-invariance implies join-closed
• Balanced implies transitive. Conversely, any function-restriction-expressible subgroup property that is transitive, must be balanced. For full proof, refer: Balanced implies transitive
• Invariance implies strongly intersection-closed. For full proof, refer: Invariance implies intersection-closed
• Left-inner implies left-extensibility-stable, that in turn implies intermediate subgroup condition. For full proof, refer: Left-extensibility-stable implies intermediate subgroup condition
• Quotient-hereditary invariance implies quotient-transitive