Virtual Morita equivalences and Brauer character bijections

We extend a theorem of Kessar and Linckelmann concerning Morita equivalences and Galois compatible bijections between Brauer characters to virtual Morita equivalences. As a corollary, we obtain that the block version of Navarro’s refinement of Alperin’s weight conjecture holds for blocks with cyclic and Klein four defect groups, blocks of symmetric and alternating groups with abelian defect groups, and p-Blocks of \(\textrm{SL}_2(q)\) ${\text{SL}}_2(q)$ and \(\textrm{GL}_2(q)\) ${\text{GL}}_2(q)$ , where p|q.