Here is a relevant section from the non-canonical books upon which the Ethiopians base their veneration of “St Pontius Pilate”
Cæsar set himself to question Pilate; and he orders a captain named Albius to cut off Pilate’s head, saying: Just as he laid hands upon the just man named Christ, in like manner also shall he fall, and not find safety.
And Pilate, going away to the place, prayed in silence, saying: Lord, do not destroy me along with the wicked Hebrews, because I would not have laid hands upon Thee, except for the nation of the lawless Jews, because they were exciting rebellion against me. But Thou knowest that I did it in ignorance. Do not then destroy me for this my sin; but remember not evil against me, O Lord, and against Thy servant Procla, who is standing with me in this the hour of my death, whom Thou didst appoint to prophesy that Thou shouldest be nailed to the cross. Do not condemn her also in my sin; but pardon us, and make us to be numbered in the portion of Thy righteous.
And, behold, when Pilate had finished his prayer, there came a voice out of the heaven, saying: All the generations and families of the nations shall count thee blessed, because under thee have been fulfilled all those things said about me by the prophets; and thou thyself shall be seen as my witness at my second appearing, when I shall judge the twelve tribes of Israel, and those that have not owned my name. And the prefect struck off the head of Pilate; and, behold, an angel of the Lord received it. And his wife Procla, seeing the angel coming and receiving his head, being filled with joy herself also, immediately gave up the ghost, and was buried along with her husband. By the will and good pleasure of our Lord Jesus Christ, to whom be the glory of the Father, and the Son, and the Holy Ghost, now and ever, and to ages of ages. Amen.