Today is an interesting feast, the feast of St. Felix. There are several things that make him interesting. First, the Felix that was usually meant was an antipope; he was illicitly installed as pope by the Emperor Constantius II while the actual pope, Liberius, was still alive. This is not unheard of; St. Hippolytus was an antipope, so it is entirely possible for an antipope to be on the calendar of saints, although it is certainly unusual. But more than that, he seems to have been an Arian bishop.
What we know, more or less for sure is that Emperor Constantius demanded that Pope Liberius condemn St. Athanasius; the Emperor supported the Arians. But Liberius did not, and he refused to cooperate. He was exiled for about two years to Thrace. We are not quite sure what happened; Liberius seems to have partly given in to the imperial demands and signed a compromise formula. This fact is why Liberius is the first pope who is not listed as a saint in the Roman calendar. It's not perfectly clear that Liberius did really give in, nor under what circumstances; and if he did, it was not by affirming Arianism but by accepting a compromise formula, i.e., one that could be read either in an orthodox way or in an Arian way. It was enough of a failure to get him left out of the Roman Martyrology and calendar, but he is on some Eastern calendars of saints, because there's no question that he himself was orthodox and he did suffer quite a bit for his orthodoxy. But in any case, during Liberius's exile, Constantius attempted to install Felix as pope. Given the circumstances, that seems to suggest that Felix was an Arian bishop, and indeed, there is some reason to think he was ordained by Arians. Constantius only recalled Liberius because the Roman population forced his hand, but once Liberius returned, Felix was forced out.
However, 'Felix' is a very common Roman name. There is at least one martyr named Felix, who seems to have been executed for opposing Arianism, and there's good reason to think that Felix II was confused with that martyr, about whom we know almost nothing else. For one thing, there's some reason to think that Felix II died in November, and at a time when there wasn't much in the way of persecution going on, so it's otherwise quite baffling why he was celebrated in July as a martyr. But the confusion, if so, goes way beyond this, because he was listed in calendars as 'Pope and Martyr'; and he was listed in lists of popes as Pope St. Felix II, successor to Pope Liberius. Rather hard on Liberius, I think, that after all his defense of orthodoxy, including two years of hard exile, he got counted as a disgrace, whereas the bishop illegally installed in his place by a heretical emperor went down in the books as a saint. Since it's pretty clear that Felix was in fact never at any time the Pope -- Liberius never resigned and continued to be regarded as the bishop of Rome both during and after his exile -- Felix II messes up the numbers; in the line of actual popes, there is a I and a III, but no II.
Nothing about this, of course, is theologically any problem; there was a Felix who was a martyr, and whose feast day has plausibly been July 29 for centuries, and nothing rides on our getting the biographies of saints right. There are certainly many examples of people becoming confused about which John, James, or Mary was being talked about in any particular context, so it's not surprising to find a case of confusion over Felix. However, there is a bit more to this. First, while Constantius did install Felix, we don't know for sure that he was actually an Arian; he could very well have been orthodox himself, despite being ordained by Arians. Even if he was Arian, he could well have ceased being so at some point. And in 1582, a grave was found claiming to hold the body of Felix, Pope and Martyr, who condemned Constantius. Gregory DiPippo discusses it here. So the fact of the matter is that we don't know, and probably never will on this side of death.
