Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles) 
A quick Question:

*

*Is there some duality known between the quasi Hopf algebra
$D^\omega(H)$ of a finite group $H$ to an orbifold model (such as
SU(2)/$G$ or SO(3)/$G$ orbifold of some group $G$)? What is this duality relation precisely?


Background:
It is known (in theoretical physics) that the algebraic framework underlying discrete H gauge theories with 2+1D Chern-Simons term is the quasi Hopf algebra $D^\omega(H)$, i.e. the Chern-Simons term introduces a 3-cocycle $\omega \in H^4(BH,\mathbb{Z}) \simeq H^4(H,\mathbb{Z}) \simeq H^3(H,U(1))$ in the cohomology group on the Hopf algebra $D(H)$. People in theoretical physics also call the quasi Hopf algebra $D^\omega(H)$ as another name: twisted quantum doubles, such as A Kitaev's (of Caltech) Annals of Physics 303, 2 (2003), Annals of Physics 321, 2 (2006). The background understanding of these topics (to me) would go to Dijkgraaf-Witten theory original paper.
My question here is inspired by the observation in this arXiv paper published in Nucl.Phys. B. It stated that: "From the point of view of conformal ﬁeld theory it is of interest to mention that the
fusion rules of $D^\omega(\mathbb{H}_8)$ for p = 1 coincide with the level 1 SU(2)/($\mathbb{Z}_2 \times \mathbb{Z}_2$)-orbifold (cited a paper by Dijkgraaf, Vafa, Verlinde, Verlinde) after
modding out the appropriate $\mathbb{Z}_2$ generated by 1 (see Table 2 here)). Apparently, the algebraic structure of such non-holomorphic orbifolds is still determined by the ‘holomorphic’ Hopf algebra, be it deformed by a non-trivial 3-cocycle. To our knowledge, this has not been noticed before."

A detailed Question:

It seems to me that there may have some duality between:
$$ \text{quasi Hopf algebra } D^\omega(\mathbb{H}_8) \text{ for p = 1} \leftrightarrow \text{level 1 SU(2)/($\mathbb{Z}_2 \times \mathbb{Z}_2$) orbifold} $$
Here $p = 1$ is the 3-cocycles labeled of $H^3(\mathbb{H}_8,U(1))=\mathbb{Z}_8$ for $p$(mod 8) in $\mathbb{Z}_8$. How about other 7 classes other than $p=1$ in $p$(mod 8)?


*

*Are there other some dualities exist for
$$D^\omega(\mathbb{H}_8) \leftrightarrow \text{? orbifold}$$
$$D^\omega(D_8) \leftrightarrow \text{? orbifold} $$
$$D^\omega(\mathbb{Z}_2^3) \leftrightarrow \text{? orbifold} $$
What is the general relation (if any, start with a finite group $H$)?
$$D^\omega(H) \leftrightarrow \text{? orbifold} $$

$D_8$ is a dihedral group with 8 group elements. $D^\omega(D_8)$ should have three labels of $p_1$,$p_2$,$p_3$ from $H^3(D_8,U(1))=\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_2$.  And $D^\omega(\mathbb{Z}_2^3)$ should have 7 labels of $p_j$ from $H^3(\mathbb{Z}_2^3,U(1))=\mathbb{Z}_2^7$.
ps. Excuse me that my mathematical background is not equivalent to a math PhD (but trained in physics), but this should be a research level question in mathematical physics. Please feel free giving comments/answers. Thank you for all who reply and support!
 A: Regarding the general question, from the point of view of conformal field theory
there is a rather trivial way to obtain (some)  $D^\omega(G)$. Namely, the representation category of the $G$-orbifold of a holomorphic (trivial representation category) rational conformal field theory is $\mathrm{Rep}(D^\omega(G))$ for some $[\omega]$, see Corollary 3.6. in 
http://arxiv.org/abs/0909.2537
But I have now idea if as Scott pointed out as a comment all finite groups $G$  (I suppose yes) can be obtained this way;  one has to find a holomorphic theory with an action of $G$, for example the Moonshine CFT for the Monster group etc. Then for a given $G$ I also have no idea which $[\omega]$ arise this way.
I think it also follows conversely, that if a CFT has $\mathrm{Rep}(D^\omega(G))$ as representation category, then it is a $G$-orbifold of a holomorphic theory.
But then I don't understand the non-holomorphic examples the op mentioned. In the non-holomorphic examples you have to mod something out to become (the dual of) $D^\omega(G)$
