F-structures and Bredon-Galois cohomology

Let F be an arbitrary family of subgroups of a group G and let Orb be the associated orbit category. We investigate interpretations of low dimensional F-Bredon cohomology of G in terms of abelian extensions of Orb. Specializing to fixed point functors as coefficients, we derive several group theoretic applications and introduce Bredon-Galois cohomology. We prove an analog of Hilbert's Theorem 90 and show that the second Bredon-Galois cohomology is a certain intersection of relative Brauer groups. As applications, we realize the relative Brauer group Br(L/K) of a finite separable non-normal extension of fields L/K as a second Bredon cohomology group and show that this approach is quite suitable for finding nonzero elements in Br(L/K).