Il Duce, for all his preening, knew very well that Italy wasn't ready for a war with the UK and France, and was also somewhat taken aback at Hitler's pact with the USSR. As a result, he sat the war out until Hitler overran France. Mussolini then entered the war, grabbing some of southern France, now that it looked certain that Hitler would win. Hitler likely knew he would do that all along, but that didn't stop him from openly sneering at Rome's sheer shamelessness. FDR and Churchill, in a moment of exceeding rareness, were at one with Berlin here.

As it turned out, Italy was more of a problem than a help for the Germans. The Italian armed forces just did not have the same kind of equipment or training that the British and French, let alone the Germans, had. Mussolini invaded Greece and British possessions in North Africa in 1940. They quickly bogged down. Mussolini's misadventures here were only turned around with German intervention. The Greeks even tried to arrange to surrender to the Germans alone, and Hitler nearly pulled that off until Mussolini figured out what was up. Mussolini, desperate to revive his prestige, committed troops to Operation Barbarossa, which ended up having to be delayed due to the aforementioned Italian miscues. This was a very, very fatal error, and moreover was a decision heavily unpopular with the Italian people, most of whom were perfectly fine with sitting out the war to begin with.

Hitler toward the end really regretted the fact that he chose to dicker with the Italians and the Vichy French rather than trying to forge an alliance with the Arab World. That's all that really needs to be said.

The Italians also managed to take British Somaliland without German support in 1940, but that didn't last too long.