Given a finite p-group P, the main result gives necessary and sufficient conditions for obtaining a torsion endo-permutation module for P by gluing a compatible family of torsion endo-permutation modules for all sections N(Q)/Q, where Q runs among non-trivial subgroups of P. This problem has applications in block theory. The paper also contains an appendix about biset functors (with a generalized Mackey formula) and an appendix about the Dade functor.