The thing that needs to be remembered is that there were technically two different Italian governments. After the Kingdom of Italy had surrendered to the allies in 1943 and switched sides. Hitler rushed a ton of reinforcements into Italy (they had known the Italians were going to surrender) in order to disarm the Italian army who was still in shock over what had happened. Once the Germans were in control of much of Italy, Hitler ordered a commando squad to rescue Mussolini from the prison he was being held in and transported back to Germany. The Germans created an Italian puppet state known as the Italian Social Republic, which was in opposition to the allied Italy in the South. So the Italians weren't technically fighting for the Germans but for the new Italian nation. Many were still loyal to Mussolini including some top generals like Rodolfo Graziani.

I've never head the claim that some were still fighting in Berlin, but one of the last units defending Berlin was an SS unit made up of French volunteers known as the SS Charlemagne.