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:

1. 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 $x$.
2. 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 $x$.
3. 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]]