It is popularly taught that the reformers declared five distinctive elements of theology that differentiated them from Rome: salvation by grace alone, through faith alone, in Christ alone, for the glory of God alone, as sufficiently taught in Scripture alone. They quoted the church fathers in support of their views to demonstrate that they weren't teaching novelties or heresies and they showed particular deference to Augustine. But did Augustine actually teach the doctrine described in the five solas?
Yes - the wikipedia article on sola fide baldly asserts that Augustine is among the "Church Fathers whom Protestant apologists believe taught the doctrine of Sola Fide (although Catholic and Orthodox apologists quote the same fathers as supporting a justification that includes works)."
Confusion in this regard, results from differing definitions of sola fide particularly between proponents and opponents. It should be correctly understood to mean: Justification comes only by grace through faith, not by any merit of works. It should not be construed so as to deny that: good works are (necessary) evidence of saving faith. Those who do so, are creating a straw man of the sola fide position, which is further clarified by:
Since Protestant apologists are in a better position to define what they mean by sola fide than their Catholic and Orthodox critics, and have adopted Augustine as a champion of the doctrine, their view is to be definitely preferred in this matter.
Another answer has (prior to a most judicious edit) profferred Augustine's On Grace and Free Will as proof positive that he does not teach sola fide, on reviewing the work however, I find it thoroughly consistent with a sola fide perspective. Consider chapters 16 & 17 of this work in their entirety, that the fair-minded may judge:
It is certainly a major counter-argument that Luther himself believed Augustine's soteriology to be deficient (evidenced in the quotes by the afore-mentioned opposing answer), an (admittedly weak) riposte is that Luther was not the only reformer to examine Augustine's work and define sola fide - other Protestant apologists evidently disagree with Luther and think Augustine's divergences from Protestant soteriology to be trivial (cf. BB Warfield's quote that was the genesis of the preceding question to this: "the Reformation was a triumph of Augustine's soteriology over his ecclesiology" and the extent to which Calvin drew from Augustine as evidenced in this Q&A).
Well, at least this appears to be Martin Luther's answer
As to your statement about the popular belief that the reformers (and their descendants) frequently quoted a plethora of Church Fathers to demonstrate the legitimacy of their doctrine...
This is simply not true. Augustine is for the most part the only Church Father cited by Luther and Calvin. The early reformers relied virtually entirely upon Scripture to formulate doctrine.
St. Augustine taught one can lose their salvation by their actions (which protestants call 'works') even if they have the faith - this contradicts the five solas and Luther's concept of Simul Iustus et Peccator (at the same time just and a sinner)
St. Augustine on how mortal and venial sins are forgiven
The Protestant Confession of Augsburg (1530) in Article 20 cites Augustine as a supporter of the Protestant doctrine of Sola Fide: “Augustine, in many volumes, defends grace and the righteousness of faith, over against the merits of works."
It would appear, however, that even today there are different opinions. Augustine, to me at least, appears to have taught what looks like a combination of the two, if not that then certainly not Sola Fide itself.
The article below (though appearing to be a personal blog, and also sources the above quote from the Confession) notes different passages Augustine wrote regarding "faith alone" AND "faith through works". http://www.willcoxson.net/faith/augprot2.htm