Did Hitler ever try to force Fascist Italy to relinquish control over South Tyrol, which had/has a sizable German population? Or was he willing to overlook the province considering his alliance with Benito Mussolini?
After the Armistice of Cassibile Nazi Germany formed the Operational Zone of the Alpine Foothills (Trentino, South Tyrol and part of Veneto) and the Operational Zone of the Adriatic Littoral (Friuli, Istria, Julian March, Slovenia), administrated by the military and de facto annexed.
Amazing coincidence, a former colleague of mine actually wrote a book on South Tyrol which covers the period you're interested in.
Georg Grote, The South Tyrol Question, 1866-2010: From National Rage to Regional State (Cultural Identity Studies) (2012). Still in print but not cheap.