Thanks for bringing up the original material.
Although not wishing to drag this thread further away from the original subject, there is an interesting article here:
The Oil Fuel Issue
By Andrea Piccinotti
After several studies, some well-known historians pointed out several discrepancies between the fuel status reports the Regia Marina was sending to the Germans and the quantity reported by the historical bureau of the Italian Navy. The most evident of these discrepancies was noted in the meeting of Merano, in February 1941, where the head of the Navy, Admiral Riccardi, stated that the Navy had only 610,000 tons left when in fact, reserves amounted to over 1 million tons. One can easily assume that the Navy had created a sort of black fund of oil fuel to be used as a last resource with the double scope of obtaining more of the now available German fuel and, in relative security, to coordinate naval operations.
The author goes on to describe the constraints on Italian Naval operations caused by the loss of Rumanian imports later in 1941. It is unclear how much oil supply constraints limited Italian participation in the German invasion of Crete in May 1941, causing the latter to rely almost exclusively on aerial invasion.
Alberto is perfectly correct about Il Duce turning up to a free lunch, his entry ticket being a few thousand dead Italians (actually only hundreds as it turned out):
On 5 June, Mussolini told Badoglio, "I only need a few thousand dead so that I can sit at the peace conference as a man who has fought".
Wikipedia: Italian invasion of France
Rather than any imagined Italian offensive naval action against Malta or wherever, French cruisers bombarded industrial sites near Genoa on 14th June 1940 and battleship Lorraine with British escorts, Bardia on 21st June. Italian naval resistance was a few torpedo boats.
Mussolini made his own separate Armistice demands of the French and it was signed in Rome on the 24th June. The demands made by Mussolini were quite limited because this was a separate agreement after the failure of the initial Italian assault in the Alps and theoretically, having concluded an Armistice with the Germans, the French could have turned all their remaining forces again him.
As I noted above, through the rest of 1940 and early 1941 there was integrated operation in the Battle of the Atlantic with the Germans U-Boat force, necessitating a free exchange of intelligence, so it is not surprising Supermarina might send some relevant decrypts to B-Dienst. Their observers would certainly have notified the Germans of the departure of Force H into the Atlantic.
All the best