Weak normal subset-conjugacy-determined subgroup

Definition
Suppose $$H \le K \le G$$. We say that $$H$$ if weak normal subset-conjugacy-determined in $$K$$ relative to $$G$$ if, for any defining ingredient::normal subsets $$A,B \subseteq H$$ such that there exists $$g \in G$$ with $$gAg^{-1} = B$$, there exists $$k \in K$$ such that $$kAk^{-1} = B$$.

The modifier weak here denotes that $$g$$ and $$k$$ may not have the same element-wise action on $$A$$.

Stronger properties

 * Weaker than::Normal subset-conjugacy-determined subgroup
 * Weaker than::Subset-conjugacy-determined subgroup

Related subgroup properties

 * WNSCDIN-subgroup: A subgroup that is weak normal subset-conjugacy-determined inside its normalizer.