  By results of the second author, a source algebra equivalence between two
$p$-blocks of finite groups induces an equivalence between the categories of
cohomological Mackey functors associated with these blocks, and a splendid
derived equivalence between two blocks induces a derived equivalence between
the corresponding categories of cohomological Mackey functors. The main result
of this paper proves a partial converse: an equivalence (resp. Rickard
equivalence) between the categories of cohomological Mackey functors of two
blocks of finite groups induces a permeable Morita (resp. derived) equivalence
between the two block algebras.
