Relative extensions and cohomology of profinite groups

We construct a correspondence between the cohomology groups of a group $G$ relative to a family of subgroups $\text{famS}$ and the classes of `relative extensions’ of $G$ by abelian groups, modulo a certain equivalence relation. We establish this correspondence for both discrete and profinite group pairs. We go on to discuss the relationships of profinite group pairs of cohomological dimension one with free products and projective group pairs.