Let $A$ be an abelian variety over $\mathbb{C}$, $L$ be a very ample line bundle on $A$, then the dual abelian variety is $\hat{A} \cong A/K(L)$ with $K(L)$ the kernel of surjective morphism $A \to Pic^0(A)$. Let $P$ be the Poincare sheaf on $A \times \hat{A}$. It is known that $\Phi_{A \to \hat{A}}^{P}$ (i.e.the Fourier-Muaki functor with kernel $P$ ) is a derived equivalence between $D^b(A)$ and $D^b(\hat{A})$.

Suppose $H \subseteq K(L)$ is a subgroup, and let $\tilde{A} = A/H$. Do we still have derived equivalence between $D^b(A)$ and $D^b(\tilde{A})$?

I feel that one could similarly define a Poincare sheaf $\tilde{P}$ on $A \times \tilde{A}$, but I am not sure if this gives the equivalence.