  Given a pair of finite groups $F, G$ and a normalized 3-cocycle $\omega$ of
$G$, where $F$ acts on $G$ as automorphisms, we consider quasi-Hopf algebras
defined as a cleft extension $\Bbbk^G_\omega\#_c\,\Bbbk F$ where $c$ denotes
some suitable cohomological data. When $F\rightarrow \overline{F}:=F/A$ is a
quotient of $F$ by a central subgroup $A$ acting trivially on $G$, we give
necessary and sufficient conditions for the existence of a surjection of
quasi-Hopf algebras and cleft extensions of the type $\Bbbk^G_\omega\#_c\,
\Bbbk F\rightarrow \Bbbk^G_\omega\#_{\overline{c}} \, \Bbbk \overline{F}$. Our
construction is particularly natural when $F=G$ acts on $G$ by conjugation, and
$\Bbbk^G_\omega\#_c \Bbbk G$ is a twisted quantum double $D^{\omega}(G)$. In
this case, we give necessary and sufficient conditions that
Rep($\Bbbk^G_\omega\#_{\overline{c}} \, \Bbbk \overline{G}$) is a modular
tensor category.
