The aim of this paper is to describe the group of endo-trivial modules for a p-group P, in terms of the obstruction group for the gluing problem of Borel–Smith functions. Explicitly, we shall prove that there is a split exact sequence  

of abelian groups where T(P) is the endo-trivial group of P, and Cb(P) is the group of Borel–Smith functions on P. As a consequence, we obtain a set of generators of the group T(P) that coincides with the relative syzygies found by Alperin. In order to prove the result, we solve gluing problems for the functor B* of super class functions, the functor * of rational class functions and the functor Cb of Borel–Smith functions.

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