While the organizational charts that I have posted look relatively clear and straightforward, the actual implementation of the chain of command was highly dependent on the men involved and their human relations and biases.
Upon the entry into war, on 11 June 1940 the Stato Maggiore Generale, which up to that date had just 6 (six!) men in its service, was expanded into the larger oranization which I have shown
here. On 13 June the undersecretary of War, gen. Ubaldo Soddu, was appointed to the new charge of Underchief of General Staff. Therefore, he was formally under the command of the Chief of General Staff, Marshal Pietro Badoglio, but at the same time he was directly in contact with the Minister of War, i.e. Mussolini himself (who was also Minister of the Navy and Minister of the Air Force). This allowed Soddu to skip the hierarchy and receive orders from Mussolini, avoiding the passage through Badoglio; a similar situation happened with the Chiefs of Staff of the Navy (adm. Domenico Cavagnari) and the Air Force (gen. Francesco Pricolo), who were Undersecretaries too.
After the death of Marshal Italo Balbo (28 June 1940), Marshal Rodolfo Graziani was appointed as new commander of SUPERASI, but at the same time he was also the Chief of Staff of the Army. Nevertheless he kept both the positions, becoming a subordinate of himself (!), as noted wittingly by his second in command, the Underchief of Staff of the Army gen. Mario Roatta. Therefore, during Graziani's stay in Libya, the true command of the Army was kept by Roatta, who limited himself to inform Graziani ex post by regular despatches summarizing his own orders and the situation in Rome.
The chain of command war made even more complicated by the Minister of Foreign Affairs, count Galeazzo Ciano, who wished to command a campaign by himself, more or less as he used to do during the Spanish Civil War. So he kept direct contacts with the commander of the troops in Albania, gen. Sebastiano Visconti Prasca, given the fact that the Lieutnant General of the King in Albania, marquis Francesco Jacomoni di San Savino, was under the direction of the Ministry of Foreign Affairs.
After the beginning of the Greek Campaign the situation became even more chaotic. In first place, on 8 November 1940 gen. Visconti Prasca was exonerated from his command and replaced, as commander of troops in Albania, by gen. Ubaldo Soddu. So Soddu was, at the same time, Undersecretary of War, Underchief of General Staff and commander in Albania. His absence from the Supreme Command was not a problem, given that Badoglio was still in Rome, but with Mussolini too busy to follow the Ministry of War, the role of Soddu was practically taken over by the chief of cabinet of the Ministry, i.e. col. Antonio Sorice, who later became a general and then Undersecretary of War (Feb.-Sept. 1943).
Soddu was replaced in his roles at the General Staff and at the Ministry by gen. Alfredo Guzzoni on 30 November 1940, therefore normalizing (at least in part) the situation in Rome.
On 5 December 1940 Badoglio was forced to resign and was replaced by gen. Ugo Cavallero as Chief of General Staff. But Cavallero immediately flew to Tirana, in order to inspect the command the Army Group of Albania, practically taking the command from Soddu's hands. Soddu was formally relieved only on 13 January 1941. So, who was really in command in Rome? During Cavallero's mission in Albania, the true Chief of General Staff was his second in command, i.e. gen. Guzzoni.
When Cavallero returned to Rome in May 1941, he suggested to Mussolini the reform which I have explained
here and his first moves were to sack gen. Guzzoni, abolishing the charge of Underchief of General Staff, and to ask Mussolini to replace him as Undersecretary with a general that Cavallero had met and appreciated in Albania, i.e. gen. Antonio Scuero. From this time on the situation of the General Staff, now always called Comando Supremo, became more normal and Cavallero managed to promote a relative cohordination among the three armed forces, despite the fact that the Chiefs of Staff of the Navy and the Air Force still kept a direct link to Mussolini given their roles as Undersecretaries. Cavallero's control on the Army, instead, was almost complete, with its Chiefs of Staff gen. Roatta and then gen. Vittorio Ambrosio.
For aeronaval operations, Cavallero was able to create also a Stato Maggiore misto per le operazioni aeronavali (Joint Staff for aeronaval operations), which comprised gen. Antonio Gandin for the Comando Supremo, adm. Giuseppe Fioravanzo for the Navy and gen. Simon Pietro Mattei for the Air Force.
Gen. Pricolo, in order to provide first-hand information to Mussolini and thus aiming to become his new military counselor, replancing the disgraced Soddu, spent most of the time between Nov. 1940 and April 1941 in Albania. Therefore, the true commander of the Air Force in this semester was gen. Giuseppe Santoro, Underchief of Staff of the Air Force. On 15 November 1941 Pricolo was replaced by gen. Rino Corso Fougier, who in turn was replaced by gen. Renato Sandalli on 27 July 1943, but gen. Santoro kept his role, guaranteeing the continuity of command in the Air Force.
The only armed force that kept an orderly and straightforward chain of command during the War was the Navy, where the Underchief of Staff was an invaluable support, especially for practical and operational matters, to the Chief, but did not replace him. The most important officer in this role, to some extent the true mind of the Italian war on the sea, was adm. Luigi Sansonetti, since 24 July 1941 till the Armistice. He succeeded adm. Inigo Campioni, who had taken this charge on 10 December 1940, when he was appointed by the new Chief of Staff of the Navy, adm. Arturo Riccardi, who in turn had replaced adm. Domenico Cavagnari (whose second in command had been adm. Odoardo Somigli).