Peter Söder, 52, and his wife, Claudia, 44, traveled to South Tyrol from the southern German state of Baden-Württemberg to see the band at Alpen Flair. “Frei.Wild isn’t just entertainment and fun, it’s also political,” Mr. Söder said. “The lyrics capture the zeitgeist in Germany,” he said.

Smoking a cigarette in the festival’s backstage area, Mr. Burger recounted how he and three acquaintances — Christian Fohrer, Jochen Gargitter and Jonas Notdurfter — formed the band in 2001, inspired by the Ramones, Guns N’ Roses and Böhse Onkelz, a German rock group that has also been accused of extreme right-wing sympathies. As a teenager, Mr. Burger sang in a neo-Nazi band called Kaiserjäger — a decision he now describes as a “mistake.” Although all of Frei.Wild’s members live in or around Brixen, Italy, a picturesque town near the Austrian border, they gradually built an audience by playing in clubs and festivals in Germany.

Mr. Burger explained that the band focused on Germany because it has a much larger market for German-language music than Italy. With their message of taking pride in one’s homeland, the band was able to appeal to nationalists in both countries — as well as in Austria and Switzerland.

The group’s success has been met with outrage from musicians, activists and politicians in Germany. In 2012, a journalist accused Frei.Wild on a TV talk show of acting like a neo-Nazi band, setting off a wider discussion about the band’s ideological position. When the band was nominated the following year for an Echo Prize, the German equivalent of the Grammys, other acts threatened to boycott the ceremony and the nomination was rescinded.