Automorphism that preserves conjugacy classes for a generating set

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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Definition

Suppose $G$ is a group and $\sigma$ is an automorphism of $G$ .We say that $\sigma$ is an automorphism that preserves conjugacy classes for a generating set if the folllowing equivalent conditions hold:

There is a generating set $S$ for $G$ such that, for all $x\in S$ , $\sigma (x)$ is in the conjugacy class of $x$ , i.e., there exists $g\in G$ such that $\sigma (x)=gxg^{-1}$ . Note that $g$ depends on $x$ .
There is a symmetric subset $T$ of $G$ that is a generating set of $G$ and such that, for all $x\in T$ , $\sigma (x)$ is in the conjugacy class of $x$ , i.e., there exists $g\in G$ such that $\sigma (x)=gxg^{-1}$ . Note that $g$ depends on $x$ .
There is a normal subset $N$ of $G$ that is a generating set of $G$ and such that, for all $x\in N$ , $\sigma (x)$ is in the conjugacy class of $x$ , i.e., there exists $g\in G$ such that $\sigma (x)=gxg^{-1}$ . Note that $g$ depends on $x$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
inner automorphism	an automorphism for which there exists a single element $g\in G$ such that the automorphism is the map $x\mapsto gxg^{-1}$	(obvious)	(via class-preserving automorphism)	\|FULL LIST, MORE INFO
locally inner automorphism	looks like an inner automorphism in its effect on any finite subset	(obvious)	(via class-preserving automorphism)	\|FULL LIST, MORE INFO
class-preserving automorphism	sends every element to within its conjugacy class	(obvious)	preserves conjugacy classes for a generating set not implies class-preserving	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
IA-automorphism	sends every element to within its coset of the derived subgroup, or equivalently, induces the identity automorphism on the abelianization	preserves conjugacy classes for a generating set implies IA	IA-automorphism\|automorphism that preserves conjugacy classes for a generating set]]