Virtual Morita equivalences and Brauer character bijections

We extend a theorem of Kessar and Linckelmann concerning Morita equivalences and Brauer character bijections between blocks to virtual Morita equivalences. As a corollary, we obtain that 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 ${\rm SL}_2(q)$ and ${\rm GL}_2(q)$, where $p|q$.