Most people think that CAG lets them do an actual PR mission just like he/she allows them to do a training TERF mission in theater, but this is incorrect. In reality, the CCDR / JFC defines a requirement for a mission he has been directed to execute (e.g. a PR requirement for an operation). Said CCDR / JFC has been allocated forces (e.g. a CSG and its airing), and if that CCDR / JFC can meet the requirements of the mission with allocated forces, then the CCDR can direct those forces to execute the mission. This assumes that the forces are manned, trained, equipped, and capable. CAG becomes a force provider to another entity who assumes TACON of those forces.
This is no different than the CFACC telling the CSG that it must provide X sorties of (insert mission here) for an operation, as has been done non-stop since 1991 in C5F.
In this construct, CAG can recommend non-supporting or stopping the support, but that kicks the decision 'upstairs' and starts a lot of important folks asking questions.
Most recent Rotor Review has a good discussion on the GFM process. There was also a good discussion of what went into this specific det on the CDRE panel @ NHA. I believe that may still be on live leak.