Questions that can be raised:
1.0. Why was there an overlap? Succession is when authority is passed from a predecessor Pontiff to a successor Pontiff? It suggests that Linus may have been proclaimed as a successor of Peter without Peter being aware.
2.0 Is he the Linus in II Timothy 4:21?
The source says;
Irenaeus also relates that Pope Linus is the same Linus mentioned by Saint Paul (2 Timothy 4:21) as joining Paul in his greetings to Timothy: “Eubulus greets you, and so do Pudens, Linus, Claudia and all the brothers.”
“Critical scholars point out, however, that Irenaeus is anything but objective in his account, the primary purpose of which is to uplift the idea of a single bishop ruling over each church, as the repository of apostolic authority. Other ancient sources indicate a collective leadership at Rome during this time, involving a group of elders, referred to either as bishops or presbyters.”
3.0. If Linus really was the successor, why did he not adopt Acts 2:38 baptism formula of Peter whereby, he used the name Jesus Christ in baptism contrary to the one being used now by Catholic which is in the name of Father, Son, Holy Spirit? Typically, a successor follows the acts of its predecessor but, in this regard, it seems Linus was not aware that Peter had used the name Jesus Christ in baptism. See here
It seems, there really is a disconnection between Peter and Linus and as such, apostolic succession as claimed may be just a CLAIM after all and not really as accurate in history.
Source: Click here