In St. Matthew’s gospel, Pontius Pilate ‘washed his hands before the multitude, saying, I am innocent of the blood of this just person’. This was to show the crowd he did not want Jesus dead, but ordered his death because that is what the people wanted. He was washing his hands of the responsibility:

Now it was the governor’s custom at the feast to release to the crowd a prisoner of their choosing. At that time they were holding a notorious prisoner named Barabbas. So when the crowd had assembled, Pilate asked them, “Which one do you want me to release to you: Barabbas, or Jesus who is called Christ?” For he knew it was out of envy that they had handed Jesus over to him. While Pilate was sitting on the judgment seat, his wife sent him this message: “Have nothing to do with that innocent man, for I have suffered terribly in a dream today because of Him.” But the chief priests and elders persuaded the crowds to ask for Barabbas and to have Jesus put to death. “Which of the two do you want me to release to you?” asked the governor. “Barabbas,” they replied. “What then should I do with Jesus who is called Christ?” Pilate asked. They all answered, “Crucify Him!” “Why?” asked Pilate. “What evil has He done?” But they shouted all the louder, “Crucify Him!” When Pilate saw that he was accomplishing nothing, but that instead a riot was breaking out, he took water and washed his hands before the crowd. “I am innocent of this man’s blood,” he said. “You bear the responsibility.” All the people answered, “His blood be on us and on our children!” So Pilate released Barabbas to them. But he had Jesus flogged, and handed Him over to be crucified.

~ Matthew 27:15-26

The Jews killed Jesus. They just used a loophole in the Roman system to do it.

And they put Jesus to death for claiming to be Yahweh. Equating oneself to be Yahweh = Blasphemy = Death. Hence why the religious leaders picked up stones to kill him multiple times throughout the Gospels.