Subgroup-conjugating automorphism

Symbol-free definition
An automorphism of a group is termed subgroup-conjugating if, under the action of this automorphism, each subgroup goes to a conjugate subgroup.

Definition with symbols
An automorphism $$\sigma$$ of a group is termed subgroup-conjugating if for any $$H \le G$$, there exists a $$g \in G$$ such that $$\sigma(H) = gHg^{-1}$$.

Alternative definitions
The following notions, permutation-extensible automorphism, and permutation-pushforwardable automorphism, turn out to be equivalent to subgroup-conjugating automorphism.

Permutation-extensible automorphism
An automorphism $$\sigma$$ of a group $$G$$ is termed a permutation-extensible automorphism if it satisfies the following:

Given any embedding of $$G$$ in the symmetric group $$\operatorname{Sym}(S)$$ over a set $$S$$, there is an inner automorphism of $$\operatorname{Sym}(S)$$ whose restriction to $$G$$ is $$\sigma$$.

Permutation-pushforwardable automorphism
An automorphism $$\sigma$$ of a group $$G$$ is termed a permutation-extensible automorphism if it satisfies the following:

Given any homomorphism $$f$$ from $$G$$ to the symmetric group $$\operatorname{Sym}(S)$$ over a set $$S$$, there exists an element $$\alpha \in \operatorname{Sym}(S)$$ such that $$c_\alpha \circ f = f \circ \sigma$$, where $$c_\alpha$$ is conjugation by $$\alpha$$.

Stronger properties

 * Weaker than::Inner automorphism:
 * Weaker than::Extensible automorphism:
 * Weaker than::Strong power automorphism

Weaker properties

 * Stronger than::Normal automorphism

Metaproperties
The subgroup-conjugating automorphisms of a group form a subgroup of its automorphism group.