Pronormal implies WNSCDIN

Verbal statement
Any pronormal subgroup of a group is a WNSCDIN-subgroup: it is a weak normal subset-conjugacy-determined subgroup inside its normalizer relative to the whole group.

Definitions used
(These definitions use the left action convention. The proof using the right action convention is the same).

Pronormal subgroup
A subgroup $$H$$ of a group $$G$$ is termed a pronormal subgroup if, for any $$g \in G$$, there exists $$k \in \langle H, gHg^{-1} \rangle$$ such that $$kHk^{-1} = gHg^{-1}$$.

WNSCDIN-subgroup
A subgroup $$H$$ of a group $$G$$ is termed a WNSCDIN-subgroup if, for any normal subsets $$A,B$$ of $$H$$, and any $$g \in G$$ such that $$gAg^{-1} = B$$, there exists $$k \in N_G(H)$$ such that $$kAk^{-1} = B$$.

Applications

 * Sylow implies WNSCDIN
 * Center of pronormal subgroup is subset-conjugacy-determined in normalizer
 * Abnormal implies WNSCC

Proof
Given: A group $$G$$, a pronormal subgroup $$H$$ of $$G$$. Normal subsets $$A,B$$ of $$H$$. An element $$g \in G$$ such that $$gAg^{-1} = B$$.

To prove: There exists $$k \in N_G(H)$$ such that $$kAk^{-1} = B$$.

Proof: Steps (5) and (6) together give the proof.