According to Article 2 of the Sardinian Statuto Albertino of 1948, which served as the Constitution of the Kingdom of Italy from 1861 through 1946, succession to the throne is hereditary according to the so-called "Salic Law." (Il Trono è ereditario secondo la legge salica.) In other words, the thousand-year old House of Savoy, which provided modern Italy its four kings, followed what is called agnatic primogeniture, by which succession is hereditary among male heirs, excluding all female descendants. The present Pretender to the vacant Italian throne, Vittorio Emmanuele di Savoia (b. 1937), who would be Vittorio Emmanuele IV if he were actually king, is the only son of Italy's last reigning king, Umberto II (1904-1983), who reigned for one month in 1946. Since only male heirs could claim the crown, Vittorio Emmanuele and later his only son, Prince Emmanuele Filiberto (b. 1972) - but not Vittorio Emmanuele's three sisters - were permanently exiled from Italy by the 1948 Constitution, until that provision was repealed in 2002.
Since the republican Constitution exiled only the two ex-Kings and their Queens (King Vittorio Emmanuele III and Queen Elena and King Umberto II and Queen Marie-Jose) and their male descendants, the exile never applied to the cadet branch of the family, descended from Vittorio Emmanuele II's second son, Amedeo, 1st Duke of Aosta (1845-1890), whose great-grandson Amedeo, the 5th Duke of Aosta (b. 1943) lives in Italy and is, in theory, the heir presumptive after Emmanuele Filiberto, who has two daughters but no son. So, if the traditional order of succession were to be followed, assuming Emmanuele Filiberto has no son in his old age, his successor as Pretender would be Amedeo, or, after him, Amedeo's own son, Prince Aimone, Duke of Apulia (b. 1967).
Meanwhile, however, Prince Vittorio Emmanuele, has decreed an amendment to the law of succession, designating his 17-year-old granddaughter Princess Vittoria, whom the NY Times describes as "a burgeoning Instagram influencer" (whatever that is) as heiress presumptive after her father Prince Emmanuele Filiberto (cf. Jason Horowitz, "Paris Teenager's New Gig: Would-Be Queen of Italy").
Unsurprisingly, the Aosta branch has objected and perhaps reasonably so, since there exists no mechanism to alter the constitutions and dynastic statutes that govern deposed royal families. In contrast, when, for example, Denmark decided to allow women to succeed to its throne in 1953, it changed its constitution, and so Denmark currently has a Queen - Margrethe II. Since then, Sweden (1980), the Netherlands (1983), Norway (1990), Belgium (1991), Denmark (2009), Luxembourg (2011) and the United Kingdom and the Commonwealth Realms (2011) have revised their succession laws even more radically, adopting what is called absolute primogeniture, according to which the oldest heir inherits regardless of sex. But deposed dynasties have no such option obviously available to them.
The senior and junior branches of the House of Savoy have not always gotten along. Famously, on May 21, 2004, following a dinner at Madrid's Zarzuela Palace on the eve of the wedding of the present King of Spain, Vittorio Emanuele punched his third cousin Amedeo in the face, causing him to fall down the steps, which reportedly caused Spain's then-King Juan Carlos to say that "never again" would he permit such an abuse of his hospitality. Then, in 2006, Amedeo declared himself to be Duke of Savoia and Head of the Royal House, claiming that Vittorio Emmanuele had forfeited his dynastic status in 1971 when he married without the authorization of his father, ex-King Umberto, something which had historically been a requirement. This has since divided Italian monarchists, some of whom still support the last king's son, Vittorio Emmanuele, others of whom have transferred their loyalty to Amedeo.
One wonders whether this latest unilateral move on the part of the senior branch of the family to alter the law of succession is just another move in that longstanding feud, one which will guarantee its continuance into future generations. Without this provocation, the feud would presumably continue into the next generation, but would then resolve itself when the males of the senior Savoia line died out, to be replaced in right order by the Savoia-Aosta branch. Such things have often happened in history. (Thus, when the Comte de Chambord died in 1883, the claim to the vacant French throne passed without qualification to the rival Orleanist branch of the Bourbon family, represented now by the Comte de Paris.)
All this squabbling, of course, can only highlight the less than illustrious modern history of an ancient and venerable dynasty, one once known, among other things, also for its piety (and its custodianship of the famous Shroud of Turin from 1453 until ex-King Umberto's death in 1983, when he willed it to the Holy See). In fact, at least two modern Savoia women are now Servants of God - Vittorio Emmanuele II's daughter, Princess Maria Clotilde (1843-1911) and Vittorio Emmanuele III's wife, Queen Elena (1873-1952). Indeed, had he had the chance, it is widely believed that Umberto II would have become an exemplary king, who would have served Italy well, perhaps better than the uninspiring Republic that replaced him. His stature, however, has not successfully been passed down to his heirs.
As a result, many, maybe most Italians may not feel much reason to miss the dynasty that not so long ago successfully created a modern and unified Italy.