  This paper introduces the notion of Brauer-friendly modules, a generalisation
of endo-p-permutation modules. A module over a block algebra OGe is said to be
Brauer-friendly if it is a direct sum of indecomposable modules with compatible
fusion-stable endopermutation sources. We obtain, for these modules, a
functorial version of Dade's slash construction, also known as
deflation-restriction. We prove that our slash functors, defined over
Brauer-friendly categories, share most of the very useful properties that are
satisfied by the Brauer functor over the category of p-permutation OGe-modules.
In particular, we give a parametrisation of indecomposable Brauer-friendly
modules, which opens the way to a complete classification whenever the
fusion-stable sources are classified. Those tools have been used to prove the
existence of a stable equivalence between non-principal blocks in the context
of a minimal counter-example to the odd Z*p-theorem.
