Meanwhile it was also running a campaign to convince the world that they'd bravely resisted the Fascists when in fact they collaborated with Mussolini for years.
I'd not heard of any particular collaboration of the Church with the Italian Fascist . The Catholic Church was bound by terms of a Concordiat to stay out of politics all together. They had to reach these agreements with any nation they operated in.
The NAZIs held the Pope a virtual prisoner inside the Vatican , a hostage to dissuade priests and laymen from actively working against them. Something like 7,000 Catholic clergy were held in concentration camps for years and hundreds were murdered by the NAZIs.
Many were beheaded for revealing that the Euthanasia programs had been expanded to get rid of disabled WW1 veterans.
Most claims of Catholic collaboration with the NAZIs are based on ignorance of the actual conditions under which any religious organization had to operate under in Europe.
An example is the "Church Tax" which was not a tax levied by the Church, it was a government tax levied to repay the Catholic Church for government use of millions of acres of Church owned property confiscated by the government for state run farms.
Its often claimed that Hitler faithfully paid his Church Tax, when in fact it was not voluntary and not paid to the church. Also since Hitler had to be granted a pardon for tax evasion, to the tune of 400,000 marks, before he could assume office its unlikely that he ever actually paid that or any other taxes.