One relevant but little noticed passage occurs in the extracts from Clement about 1 Peter made by Cassiodorus.
The idea of Peter preaching to the equites or knights of Caesar in Rome is rather striking."Marcus, my son, salutes you." Mark, the follower of Peter, while Peter publicly preached the Gospel at Rome before some of Cæsar's equites, and adduced many testimonies to Christ, in order that thereby they might be able to commit to memory what was spoken, of what was spoken by Peter, wrote entirely what is called the Gospel according to Mark. As Luke also may be recognised by the style, both to have composed the Acts of the Apostles, and to have translated Paul's Epistle to the Hebrews.
The only early parallels I can find occur in the Acts of Peter
And Dionysius and Balbus from Asia, Roman Knights, illustrious men, together with a senator named Demetrius, took a hold of Paul's right hand and said, "Paul, I should like to leave the city and be always with thee, if Iwere not a state's officer." In like manner spoke Cleobius and Iphitus and Lysimachus and Aristeus of the house of Caesar...
And on Sunday Peter spoke to the brethren and encouraged them in the belief on Christ. And many senators and knights and wealthy women (and) matrons were present...
The simplest expanation is that Clement was familiar with something like our Acts of Peter