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 characteristic subgroup	End. $\to$ End.	Yes	Yes	Yes	No	Yes
I-characteristic 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	Quotientable $\to$ Quotientable, Inner $\to$ Quotientable	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	No	Yes