Fixed-point subgroup of a subgroup of the automorphism group

Definition
A subgroup $$H$$ of a group $$G$$ is termed a fixed-point subgroup of a subgroup of the automorphism group if there is a subgroup $$B$$ of the automorphism group $$\operatorname{Aut}(G)$$ such that $$H$$ is precisely the subset of $$G$$ fixed under the action of $$B$$.

Formalisms
This property is the property of being closed with respect to a Galois correspondence between the group and its automorphism group, via the fixed point relation.