Function restriction formalism chart

Subgroup property	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