# Function restriction formalism chart

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