Abelian and pronormal implies SCDIN

Verbal statement
An abelian pronormal subgroup (i.e., a fact about::pronormal subgroup that is also an abelian group) of a group is a SCDIN-subgroup: it is subset-conjugacy-determined in its normalizer relative to the whole group.

Statement with symbols
Suppose $$H$$ is an abelian pronormal subgroup of a group $$G$$. Then, $$H$$ is a SCDIN-subgroup of $$G$$: given any two subsets $$A,B$$ of $$H$$ that are conjugate by $$g$$ in $$G$$, they are conjugate by some $$h$$ in $$N_G(H)$$, and the action of $$h$$ coincides with the action of $$g$$ on $$A$$.

Facts used

 * 1) uses::Pronormal implies MWNSCDIN
 * 2) uses::Abelian and MWNSCDIN implies SCDIN

Proof
The proof follows directly from facts (1) and (2).