Although Jesus was the one who hung upon the cross and took the punishment for our sins, the work of atonement was a combined effort of the Father, Son and Holy Spirit. The gospels do not specifically mention the Father or Holy Spirit, other than Jesus’ cries to his Father on the cross (Abba). But looking closer, the Father and Holy Spirit can be found behind the scenes making sure every step went according to plan.
About the practice of crucifixion, James Jeffers notes that by the first century, crucifixion had already been a practice of execution for quite a while. In his book, “The Greco-Roman World of the New Testament Era,” he mentions that Alexander the Great had used this form of torture, and it is possible the Romans learned it from the Carthaginians of North Africa.
This form of execution was designed to be slow and painful. The condemned person’s weight was supported for the most part by his arms. Muscle spasms, cramps and insects added to the pain, and death usually came through gradual suffocation as blood entered the lungs. Romans sometimes broke the legs to increase the weight and bring death more quickly.
During his ministry, Jesus played an active role in fulfilling prophecy. But at the point of crucifixion Jesus could do nothing more than carry his cross (Matthew records Simon of Cyrene at one point taking Jesus’ cross) and be hung there. Everything else regarding prophecy was done to him, and he had to rely solely on his Father and Holy Spirit to make sure everything around him was happening as planned.
To say which entity—the Father or the Holy Spirit—did which unseen actions is too speculative. But it can be said that the other triune entities made sure that prophecy was being fulfilled in the casting of lots for Jesus’ garment; that Jesus died before the soldiers came to break his legs, that his side was pierced, and others.
It is probable that it was the Holy Spirit that gave revelation of Jesus’ innocence and identity to the Roman soldier; and it was likely the Father who brought about the darkness, the earthquake and perhaps rent the temple veil in two. It can be said for sure that it was the Father who accepted his son’s sacrifice (who looked away as he became sin), who decided whether or not to forgive the mockers (Luke 23: 34) and carried away his son’s spirit (Luke 23:46) into Sheol or Hades (the grave).
While we have almost exclusively given credit to Jesus for what was done on the cross, the crucifixion was perfectly accomplished by all three members of the Trinity. This was His (their) effort to come to us, to do what was impossible for us to do for ourselves—to make us holy in the eyes of the Lord (John 3:16-17).
 H. Wayne House, Chronological and Background Charts of the New Testament (Grand Rapids, MI: Zondervan), 28-36.