Let $G$ be a finite group, with subgroups $A \leqslant H$. Is there an isomorphism of $N\_G A$-sets (or just sets)
$$ N\_G A / N\_H A \cong (G/H)^A ?$$


This dropped out of some calculations of Mackey functors, but I would like a more direct proof (or a counterexample).



Quotient of normalizers is the fixed points of a homogeneous space
The statement is false, as shown by the counter examples of LSpice and Neil Strickland.


