The session of the Grand Council of Fascism where Mr. Grandi's motion to have Mussolini step down was carried, was held on July 24th 1943, which was the middle of summer. Everyone in the fiction is instead dressed as if it is winter time, both the soldiers and the crowd cheering in the streets.
Bruno Mussolini's real life air crash took place in August 1941, well after the Italians entered WW2, not in 1934 as depicted in the series. Bruno also fought in the Spanish Civil War, the first major alliance between Hitler and Mussolini, which is strangely omitted from this series.