Vitellius was governor of Syria and, therefore, Pontius Pilate's immediate supervisor. He did dismiss Pilate in 36 A.D.
I offer this explication of the fifth through fourteenth lines: As Emperor, Tiberius was a stern and demanding administrator, and expected complete compliance with imperial administrative and operative policies. One of these is that administrators of unsettled provinces, where the presence of legions were required to enforce Roman law (and especially the tax collection), were not permitted to bring their immediate families into those provinces. Yet, Matthew's Gospel is quite clear in chapter 27 that Pilate's wife was with him not only in the provincial capitol of Caesarea, but also when he went up to Jerusalem to enforce the Pax Romana during the Passover Feast. This was a blatant disregard of Tiberius' policy, and yet the Emperor did not act against Pilate for this particular violation. Given that Pilate was not a particularly influential Roman, one must seek an explanation elsewhere.
In his adolescence, Tiberius, the stepson of the first Emperor, Augustus, married Vipsania Agrippina. and accounts indicate that their marriage was fervently happy. At the time Tiberius' two main ambitions were to become a scholar of Greek Poetry (an interest he maintained into his old age) and to make babies with Vipsania. Apparently, they practiced making babies constantly. Vipsania, her father's eldest child, had been raised frugally. She is said to have done her own marketing, to have walked around outdoors barefoot, and to employ only one or two servants; she also is said to have prepared their meals herself. When her father, who was the first Emperor Augustus' chief administrator died, wealthier than Augustus himself, Vipsania, as his eldest child, received half of the estate (the other half to be divided among her numerous siblings). With this sum, she could provide her husband with the means to travel in the East to study Greek Poetry. At this point, she became a threat to the plans of Tiberius' mother, Livia, who hoped to see her son installed eventually as the second emperor, although Tiberius was said not to share that ambition. Through manipulation, Livia convinced Augutus to compel Tiberius and Vipsania to divorce (the news of which caused her to miscarry their second pregnancy); Augustus also compelled Vipsania to marry a Senator, Asinius Gallus. When Augustus died, Tiberius ordered Livia, his mother, to vacate the imperial palace and never spoke to her again. It is said that when she passed away, he did not attend her funeral but went to his office and worked through the day as usual.
Vipsania passed away in 20 A.D. She was survived by several children and grandchildren, and one of these was her grandchild, Asinia. At the funeral, Tiberius is supposed to have approached Asinia to tell her that she looked exactly as her grandmother had looked when they married, and then he walked away and returned to the imperial palace. Tiberius had more than enough authority to compel Asinia---had he so desired---to become his concubine. I believe he was aware of this temptation, knew that he would not be able to long resist it, and took the most effective solution to his problem: to marry her to someone whose work would take her far, far from Rome, and far out of his immediate reach Marriage also provided further insulation for her from Tiberius---as he was the supreme pontiff of pagan Rome, its chief religious authority, and to have interfered in her marriage would have been a most unseemly action. I believe that he married her to Pontius Pilate. Lloyd Douglas, in his novel The Robe, mentions that Asinia had a very persuasive power over Tiberius---and, building on that assertion, I suggest that she chose to go up to Jersualem with her husband for the Passover, because Tiberius would not have denied her decision, and during that time she advised Pilate not to become involved with the plot to murder Jesus.
