Did the Savoy's have a good or bad relationship with Mussolini and Hitler?
Italy remained nominally a monarchy throughout the Mussolini era and, under the constitution of the Kingdom of Italy, King Victor Emmanuel III could have in theory dismissed Mussolini as head of government at any time. You could argue that, in practice, that was not necessarily the case as any attempt to remove Mussolini could have pushed the dictator to abolish the monarchy itself. There were, however, several moments in time, in the 1920s and 1930s, when Mussolini was in a weak position and the King could have realistically removed him with the backing of the Italian military and several political factions. The King, nevertheless, failed to do so. Moreover, King Victor Emmanuel was directly responsible for enabling Mussolini to come to power in the first place as he refused to call a state of emergency to suppress the Fascists' March on Rome and then called upon Mussolini to form the government. The King also remained silent when Mussolini de facto censored the press, got rid of the opposition parties and replaced free and competitive multiparty elections with plebiscitary one-party lists.
Did the King sympathize with fascism and favor it over the liberal parliamentary democracy that existed in Italy before Mussolini came to power? Or was he concerned only with the survival of his dynasty and his reign above the national interest? I am no expert, but I would say that both are probably true. Victor Emmanuel was also clearly supportive of Italian imperialism under Mussolini and I believe greatly enjoyed his own upgrade to the status of "King Emperor" on par with other European monarchs.
Eventually the King turned on Mussolini and deposed him (when the Anglo-American allies were already on Italian soil and the direction the war would take was clear), but it was probably too little too late. He clung to Mussolini for over 20 years as a safeguard against communists, socialists and republicans , and , ironically, by doing so, ultimately all but guaranteed the demise of the Italian monarchy and, even more so, of the House of Savoy.
