  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 $p|q$.
