= Block induction requires the block's defect centralizer
The centralizer condition defining <block induction> refers to a <defect group of a block> of the actual source block. Containment of the centralizer of an unrelated <p-subgroup> does not suffice. For example, in characteristic $3$ take $G=A_5$, $H=A_4$ and $D=C_3$. Then $DC_G(D)\le H$, but the unique defect-zero block of $kA_4$ occurs on bimodule restriction of both three-dimensional defect-zero blocks of $kA_5$. The two ordinary degree-three characters have identical restriction to $A_4$, namely its degree-three character, and defect-zero reduction gives the same simple projective module. Thus the unique-block characterization fails if $D$ is not the defect group of the source block.
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