What Dai Grepher is saying is somewhat borne out by the storyline but I still think that what PW says is correct and the leadership in Par Vollen had authorised her mission.
Essentially the moment the big hole appeared in the sky, the Qun leadership decided they needed to do something about the out of control mages in the south. This makes sense since this large scale reality changing magic affects everyone. They took a two pronged approach, authorising the Viddasala to carry out her own investigations and ordering Hissrad to ingratiate himself with the Inquisition.
What is less clear is at what point the Viddasala's actions went off brief. We know from that Templar convert in the Deep Roads that her actions with the Saarebas were beyond what she should have been doing, which I believe Bull confirms, but essentially what she was trying to do was find a way of strengthening the Veil and reducing the flow of magic, which had she succeeded would have been something the leadership would have welcomed, so who is to say that she hadn't contacted Par Vollen, got their authorisation but with the proviso, if it goes wrong then we will disavow all knowledge.
The same would hold true of the planned invasion. Obviously there would be no point her going ahead with the plan without the leadership being in the know because the whole point was to take advantage of the chaos following the assassination of the leaders across the south. So their agents in the cities would need to be primed ready to act when the moment occurred, as would their forces (using the eluvians for fast travel from Par Vollen). They wouldn't have needed huge forces in each city; the Arishok nearly succeeded in taking over Kirkwall with only a token army of qunari and the support of their elven converts. All they needed was the signal to act; the large explosion in the seat of government. However, when the plot was uncovered at the eleventh hour they naturally had to disavow their involvement both to safeguard their agents and the treaty that is keeping the south from aiding Tevinter against them. It is noticeable in the epilogue that if you sided with the Qun, they contact the Divine and ask for her support (and that of her private army the Inquisition if you keep it going) against Tevinter, again insisting that the Viddasala was acting on her own.
The only truth that the Qun are loyal to is the Qun itself. Members of the Qun, particularly those higher up, should be well aware that they are dispensable if they fail in the mission, since they have not fulfilled he demands of the Qun, one of the chief ones is not to fail. Whilst the Arishok may have recovered the tome, his mission was to do so without compromising the peace treaty with the southern nations; had his takeover of Kirkwall been successful, then the leadership would probably have overlooked this aspect. However, it was not successful, so even if he returned home with the tome, he was always going to be stripped of office and the leadership disavow his actions. The same is true of the Viddasala; she attacked Solas likely knowing what he would do to her because she had failed and placed them in a compromising situation, so even had she returned home, that would have been an end of her role and likely her life.
As for Hissrad, he was always a conflicted character but being lower on the pecking order, as someone loyal to the Qun he was still required to follow the orders of a superior. I could be wrong here but I believe the investigation of magic was a branch of the Ben-Hassrath, so she was the same division as him. In any case she was definitely an officer whilst he was a mere agent. It is one of the weaknesses of the Qun that their members are meant to blindly follow orders. Hissrad did that when you ordered him to sacrifice the Chargers, so why should he be any different when ordered by the Viddasala to kill you? Whilst the Qun may have wanted to salvage the peace treaty, it would still serve their interests to get rid of the Inquisitor, so the secrets that were discovered in the Viddasala's base would never be known.