Conjugacy classes and characters for extensions of finite groups

Let $H$ be an extension of a finite group $Q$ by a finite group $G$. Inspired by the results of duality theorems for étale gerbes on orbifolds, we describe the number of conjugacy classes of $H$ that maps to the same conjugacy class of $Q$. Furthermore, we prove a generalization of the orthogonality relation between characters of $G$.

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