Blocks of central $p$-group extensions

Abstract:

Let $G$ and $G’$ be finite groups that have a common central $p$-subgroup $Z$ for a prime number $p$, and let ${\overline {A}}$ and ${\overline {A’}}$ respectively be $p$-blocks of $G/Z$ and $G’/Z$ induced by $p$-blocks $A$ and $A’$ respectively of $G$ and $G’$, both of which have the same defect group. We prove that if ${\overline {A}}$ and ${\overline {A’}}$ are Morita equivalent via a certain special $({\overline {A}}, {\overline {A’}})$-bimodule, then such a Morita equivalence lifts to a Morita equivalence between $A$ and $A’$.