Modern scholarship has long rejected Petrian authorship. Conservatives are less likely to accept that verdict but essentially there seems to be no strong compelling reason to accept Peter as author of I Peter. There may,however, be a couple of reasons to question that conclusion.The assumption among modern scholars is against Petrine authorship, but there are some arguments for it. First the weak:
-attributes itself to Peter
-knew christians from far away who Peter could have met on Pentecost
-Claims to have been sent from Babylon; babylon means Rome;we know that Pete wound up in Rome documented by 1 Clement 
-The author of the letter also indicates that he has a close relationship with “Marcus my son” (1 Peter 5:13). This may be the same John Mark with whose family Peter had found refuge years earlier.
Those may be good answers in context but anyone could say those things That does not prove Peter wrote the epistle. The best reasons to think he did are that the early fathers understood that he did. The second reason is the above context and the realization that the author gives away his Jewishness in the use of Hebrewisms related to speaking of God. Chapter 2:11. Sets himself and readers apart from Gentile pagans.
For my money the best reason to deny Petrian authorship is because the book is so well written.  It doesn't seem likely that a Galaleian fisherman would know such fine Greek. Since Papias says he used John Mark as interritor in Rome it seems clear he could not have written in Greek.But he could have asked someone to fix the letter for him. So it could be that the letter is based upon Peter's ideas but brought to reality by words of another. Maybe that's what he means by saying Silas helped with the letter at end of chpatr 5,v"12 With the help of Silas, whom I regard as a faithful brother, I have written to you briefly, encouraging you and testifying that this is the true grace of God. Stand fast in it."
It is also possible that John Mark was the one who wrote the letter for him. Of course we could never prove that.
This raises a question about Marcan authorship of the gospel of Mark. If Mark was interpreter for Peter in Rome we should expect him to communicate well in Greek. Yet the Gospel of Mark is not well written. My Greek professor said his Greek was horrible and his professor called Mark "the illiterate one." He gets tenses wrong, he gets persons; wrong as in first and second and many other things.
"Ben Witherington in The Gospel of Mark: A Socio-Rhetorical Commentary (pp. 18-9) documents a number of stylistic traits of Mark's Gospel:"
Historical present tense verbsAs Witherigton Puts it:
Repetition of phrases
Impersonal plural verb followed by a singular verb
First-person plural narrative
Unusual words or constructions
"In sum, these traits point to an author who struggles to express himself in the language he is writing... So the text itself suggests the author of Mark was, in fact, an Aramaic speaker."
Mark could still stand behind the document as its root source although it was written by someone else. Mark could have imparted the knowledge Peter gave him to the community and the community produced the actual author. The name Mark is associated with the work because it was a product of the Mark community.
Neither case can be proven and it may be likely either namesake stands behind the work. Does that mean the works should just be chucked out? Apostoloic authorship was a major criterion for acceptance in the cannon, That is the reason they kept sticking names of prominent evangelists on Gospels. Even so in these documents the Bishops saw the rudiments of the Gospel as they were meant to be. Thus these books belong in the canon.
Michael J. Kok "Peter in Rome: Peter endured many labours in 1 Clement 5:4," The Jesus Memoirs... ) (June 9:2017)
Kok: Current Position: New Testament Lecturer and Dean of Student Life, Morling College Perth Campus
Education: Bachelor of Arts in Religion and Theology with a Specialization in Biblical Studies (Taylor University College), Master of Arts in Religious Studies (University of Alberta), Ph.D. in Biblical Studies (University of Sheffield)
1 Clememt 5:4 documents Peter in Rome, he was present himself and saw Peter.
for Text of Clement see Peter Kirby, early Christiam writtings: https://www.earlychristianwritings.com/text/1clement-lightfoot.html Also backed by St Ignatious. Two greats attest to it.
Persuasive evidence does exist that Clement had personal contact with Simon Peter and studied under the apostles. Irenaeus of Lyons (AD 130—200) informs us that “this man [Clement of Rome], as he had seen the blessed apostles, and had been conversant with them, might be said to have the preaching of the apostles still echoing [in his ears], and their traditions before his eyes” (Against Heresies 3:3).https://www.gotquestions.org/Clement-of-Rome.html
Frank F. Judd Jr.,"The Case for Petrine Authorship of 1 Peter," The Ministry of Peter the Cheif Apostle, Frank Judd, Religious Studies Center,BYU,(bi date listed) https://rsc.byu.edu/ministry-peter-chief-apostle/case-petrine-authorship-1-peter
Ben Witherington, in "What Evidence Suggests That Greek was NOt Mark's First Lamgague," Biblical Hermeneutics, Stack Exchage. (Jyly 30, 2018) https://hermeneutics.stackexchange.com/questions/2021/what-evidence-suggests-that-greek-was-not-marks-first-language