Clifford theory of characters in induced blocks

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 $b'$ of $N$ and ${\rm N}_N(D)$, respectively, with $(b')^N=b$, where $D$ is a defect group of $b'$. 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 $\phi^G$ and those of $(\phi')^{{\rm N}_G(D)}$ where $\phi$ and $\phi'$ are irreducible Brauer characters in $b$ and $b'$, 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.