Normal Hopf subalgebras in cocycle deformations of finite groups

Let $G$ be a finite group and let $\pi :G\to G^{\mathrm{\prime }}$ be a surjective group homomorphism. Consider the cocycle deformation $L=H^{\sigma }$ of the Hopf algebra $H=k^G$ of $k$-valued linear functions on $G$, with respect to some convolution invertible 2-cocycle $\sigma$. The (normal) Hopf subalgebra $k^{G’} \subseteq k^G$correspondstoaHopfsubalgebra$L’ \subseteq L$. Our main result is an explicit necessary and sufficient condition for the normality of $L^{\mathrm{\prime }}$ in $L$.