Quotients of strongly graded quasi-Hopf algebras

Let $H$ be a strongly $G$-graded quasi-Hopf algebra, with $G$ an arbitrary group. We classify its quasi-Hopf ideals by showing that each quasi-Hopf ideal $I$ of $H$ corresponds to a pair $(N,K)$, where $N$ is a normal subgroup of $G$ and $K$ is a $G\mathrm{/}N$-invariant quasi-Hopf ideal of $H_N={⨁}_{n\in N}{H}_n$ whose image under the grading map contains the augmentation ideal of $\mathbb{k}[N]$. We further improve this correspondence for particular classes of quasi-Hopf ideals, such as those that have an associated quasi-Hopf retraction, or those whose lifted quotient is a crossed product. Applications include cocentral abelian extensions with infinite grading, finite neutral component and nontrivial reassociator.