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 $Frightarrow overline{F}:=F/A$ 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 $Bbbk^G_omega#_c, Bbbk Frightarrow Bbbk^G_omega#_{overline{c}} , Bbbk overline{F}$. 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.
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