Local conjugacy and primary-type decompositions in nonabelian cohomology

In 1979, Losey and Stonehewer (Warwick) introduced local conjugacy and provided some conditions under which locally conjugate complements J & J' to a normal nilpotent subgroup N must be conjugate. We will interpret their results in the language of nonabelian group cohomology and relate them to the existence of a primary-type decomposition of the first cohomology set H^1(J,N). We will cover the cases: N is abelian, J is nilpotent, and N⋊J is supersoluble. This decomposition will also provide us with an inclusion-based version of their result. We will motivate our discussion in terms of fixed point results in the style of Glauberman (Chicago).