Did the Fascist party kill Jews and communists like the Nazis did?

The term "Fascist Party" is not unique to one country, but is a descriptor of an ideological party that can be found in multiple countries.

If we are talking about the Italian Fascist Party, then, yes, they did. While the Italians did not actively contribute to the financial cost and development of the Holocaust, Italy and Italian possessions ghettoized their Jews and later sent them to the Death Camps and imprisoned Communists.