The Brauer indecomposability of Scott modules is important for obtaining splendid Morita equivalences between the principal blocks of two finite groups whose fusion systems on their Sylow p-subgroups are isomorphic. In this talk, first the connection between the Brauer indecomposability of a p-permutation module and the saturation of the corresponding fusion system will be discussed. Then, recent results along these lines for some special families of 2-groups, including semidihedral and wreathed 2-groups, will be presented. Part of this work is joint with S. Koshitani.