Augustine was not steadfast in his interpretation of Matthew 16:18. His quotes below show that Augustine equated the rock with Peter's faith, Peter's successors, and Peter himself. It was during his controversies with the Manicheans, Donatists, and Pelagians that he emphasized the role of Christ and identified "this rock" with Christ. In his dealings with the Manicheans, the nature of God was in the forefront; with the Donatist, it was the nature of the Church and clergy; with the Pelagians, it was the nature of grace and its originator, Jesus Christ. Augustine equated "this rock" with Christ not to downplay Peter's primacy, rather to emphasize Jesus Christ. Against all these heresies, Augustine stressed that the Church's foundation and grace rested upon a divine and not a human person. Nevertheless, Augustine remained steadfast in his understanding of Peter's primacy and the primacy of the Roman See. Augustine did not reject the Petrine interpretation, in favor of which he cites Ambrose's hymn, but leaves it to the reader to choose. Simon remains a rock, a secondary rock dependent on the Rock-Christ, for Augustine writes, 'Peter having been named after this rock '
Here are several citations from Augustine to show that he had a well developed understanding of the primacy of St. Peter and of his successors, the bishops of Rome. Let's take a look at St. Augustine's treatment of Matt 16:18 a little closer.
"Number the bishops from the see of Peter itself. And in that order of Fathers see who succeeded whom, That is the rock against which the gates of hell do not prevail."
Psalmus contra partem Donati, 18 (A.D. 393),GCC 51
"Let us not listen to those who deny that the Church of God is able to forgive all sins. They are wretched indeed, because they do not recognize in Peter the rock and they refuse to believe that the keys of heaven, lost from their own hands, have been given to the Church."
Christian Combat, 31:33(A.D. 397), in JUR,3:51
"For if the lineal succession of bishops is to be taken into account, with how much more certainty and benefit to the Church do we reckon back till we reach Peter himself, to whom, as bearing in a figure the whole Church, the Lord said: 'Upon this rock will I build my Church, and the gates of hell shall not prevail against it !' The successor of Peter was Linus, and his successors in unbroken continuity were these: -- Clement, Anacletus, ... In this order of succession no Donatist bishop is found."
To Generosus, Epistle 53:2(A.D. 400), in NPNF1,I:298
"When, therefore, He had said to His disciples, 'Will ye also go away?" Peter, that Rock, answered with the voice of all, "Lord, to whom shall we go? Thou hast the words of eternal life.' "
Homilies on John, Tract 11:5(A.D. 417), in NPNF1,VII:76
"And the Lord, to him to whom a little before He had said, 'Blessed thou art, and upon this Rock I will build my Church,' saith, 'Go back behind, Satan, an offence thou art to Me.' Why therefore 'Satan' is he, that a little before was 'blessed,' and a 'Rock' ?"
In Psalms, 56[55]:14[PL 36, 656] (A.D. 418),in NPNF1,VIII:223
"Peter, who had confessed Him as the Son of God, and in that confession had been called the rock upon which the Church should be built."
In Psalms, 69:4[PL 36, 869] (A.D. 418), in Butler, 251
"And if a Jew asks us why we do that, we sound from the rock, we say, This Peter did, this Paul did: from the midst of the rocks we give our voice. But that rock, Peter himself, that great mountain, when he prayed and saw that vision, was watered from above."
In Psalms, 104[103]:16(A.D. 418),in NPNF1,VIII:513
To say that the Church is built on Christ would not be against Catholic teaching. However, a proper interpretation of Mt 16:18 leads one to conclude Peter is the rock.
Jesus spoke in Aramaic. We know this because he says on the cross in Mt 27:46, "Eli, Eli, lama sabachthani?" This is Aramaic, not Greek.
In addition, Paul writes in Galations 4 times and in 1 Corinthians 4 times, Simon's new name in Aramaic. Our English bibles translate this as Cephas which is a transliteration of the Aramaic work Kepha.
And, Kepha means 'Rock'. It means the same as 'petra'. What Jesus said to Simon in Matthew 16:18 was this: ‘You are Kepha, and on this kepha I will build my Church.’
When you see what Jesus said in the Aramaic you understand that he was equating Simon as the Rock and that he would therefore build his Church on him signified by changing his name to Kepha and saying the words 'upon this Kepha I will build my church'.
In Aramaic you can use kepha in both places in Matthew 16:18. When translated into Greek you encounter a problem arising from the fact that nouns take differing gender endings. You have masculine, feminine, and neuter nouns. The Greek word petra is feminine. You can use it in the second half of Matthew 16:18 without any trouble.