Clifford theory of characters in induced blocks

Abstract: We present a new criterion to predict if a character of a finite group extends. Let $G$ be a finite group and $p$ a prime. For $N\lhd G$, we consider $p$-blocks $b$ and $b&#39;$ of $N$ and ${\rm N}_N(D)$, respectively, with $(b&#39;)<sup>N=b$, where $D$ is a defect group of $b&#39;$. Under the assumption that $G$ coincides with a normal subgroup $G[b]$ of $G$, which was introduced by Dade early 1970's, we give a character correspondence between the sets of all irreducible constituents of $\phi<sup>G$ and those of $(\phi&#39;)^{{\rm N}_G(D)}$ where $\phi$ and $\phi&#39;$ are irreducible Brauer characters in $b$ and $b&#39;$, respectively. This implies a sort of generalization of the theorem of Harris-Kn\"orr. An important tool is the existence of certain extensions that also helps in checking the inductive Alperin-McKay and inductive Blockwise Alperin Weight conditions, due to the second author.