Let's think about the conspiracy to murder Jesus Christ in the context of the Roman Empire's politics.
I believe this occurred on the weekend of April 3, 33 (and, there being no Year 0, Jesus was born in Fall, 1 BC).
Tiberius withdrew to Capri in 26. Sejanus was to administer the entire Roman Empire. He was the most powerful man, unless Tiberius decided he wasn't. Sejanus appointed Pilate governor of Judea.
So Pilate was of the Sejanus group, not necessarily of Tiberius. Sejanus was executed by Tiberius in 31 because Sejanus was widely suspected to be a conspirator against the Emperor.
"If thou release this man, thou art not Caesar's friend. For whosoever maketh himself a king, speaketh against Caesar." John 19:12
This is a clear threat against Pilate, an appointee of Sejanus and his men. The crowd would denounce him to Tiberius as disloyal. "Not Caesar's friend." Pilate would be crucified, probably along a main Roman road as a public warning, and his entire family exiled or murdered - including his wife Saint Procla, who joined St. John and the Holy Women that very April weekend.