In early 1944, Giovanni Preziosi - a former priest and notorious Italian antisemite - wrote to Mussolini:

The first step is not that of creating a national constituent of the State, but that of eliminating the Jews. As often throughout history, the capital adversity is not that of shaping the new state of things, but that of making room for it. […] Our first central task is not the so called national unity […] but the total elimination of the Jews [including] rooting out those more or less baptized, arianized. Then push out of any vital center of the national life […] the mongrels, husbands of Jew women and anyone with a single drop of Jewish blood.

Preziosi's argument echoed in large part (and intentionally) Hitler's own take on the Jewish question. The solution of the difficulties of the Fascist Regime during the war, the internal opposition culminating in Mussolini's removal from power in July 1943, was inextricably tied to a definitive solution of the Jewish problem. Those events had been the result of a "Jewish conspiracy", as Preziosi had already explained in a few contributions writted for the Volkischer Beobachter - Preziosi's choice of words was, again, not accidental: this private letter, with an appeal to Mussolini not to focus too much on the reorganization of the puppet Repubblica Sociale but to concentrate his efforts on the most urgent issue, was not supposed to remain private.

Preziosi did not in fact expect a positive answer from Mussolini - Mussolini understood more or less that there was no urgent "Jewish question" in Italy, if only for numerical reasons, and had been already satisified with the first steps taken after September 1943 when the Jews had been downgraded to "enemies of the State" [Those belonging to the Jewish race are foreigners. During war time they belong to an enemy nation explained the manifesto of Verona, the "founding chart" of the new state] and subject to various measures of requisitions and concentration.

This had opened the phase of "economical persecution" of the Italian Jews, with the Regime focused on collecting and seizing properties large or small of the few Jews left under their control.

Preziosi - who in this regard was more or less an outlier in the Italian political world, like many Nazi collaborators, and was openy despised by Mussolini - had made sure that the German authorities had been informed of his request, so that Mussolini's dismissive answer would have looked as new evidence of lack of resolve on the matter.

Preziosi's plan appears to have worked, as he was placed in charge of the deportation of the Italian Jews on March 15^th 1944. He would then oversee the process up to the point where the Italian Jews were given up to the Germans.

While Preziosi's letter is not evidence that he necessarily knew the inner workings of the Holocaust machinery, it's apparent at least that he aimed his efforts at the "elimination of the Jews". And that there were therefore groups within the Italian puppet Regime which operated organically (and intentionally) to the general system.

More broadly speaking, there was little focus on the Jews though - and the Germans were extremely private with that kind of informations (they were generally not inclined to share informations with the Italians unless they needed to), so that inquiries made by the Italian authorities were rapidly dismissed or even met with contempt and threats.

Consider for instance the answer given by the German IVB4 to the RSI foreign office inquiring about one deported Jew:

We do not wish to move any further with the Italian Embassy's request to make inquiries about the actual location of a Jew Taubert. It is our best advice for the Italian diplomacy never to make such requests again. We need time and hard work to attend to much more vital things for the general interest. […] We are deeply concerned that the Embassy of a fascist republic continues to employ its influence in favor of the Jews.

This excerpt should provide some evidence that not every group (by 1944 there was a lot of confusion within the puppet regime, and its bodies worked often in a rather uncoordinated manner, so that personal initiatives from functionaries could work in totally contrasting directions) within the Salò Republic had a clear picture of the fate of the deported Jews - and that there was nonetheless concern over the handling of what many continued to see as Italian citizens.

As for Mussolini himself - he was informed, at least in broad terms, of the German conduct of war in the Eastern territories. He did not appear to take much concern with the technical workings of the process, or with the racial cathegorizations. Consider anyways that Mussolini had met repeatedly with Hitler, that he had visited the Eastern front (albeit far from the actual front as far as I know), that the Italian forces had taken an active part in the Russian campaign in the region north-west of Stalingrad. More so, the Italians were directly involved as an occupation force in Yugoslavia, where they operated along side with the Croatian Regime and the German occupation forces.

The Italian authorities had often taken preemptive measures with the troops coming back, to avoid them spreading news of certain things they may have seen. The "poor" - to use a substantial understatement - treatment of the populations was certainly apparent to those who had witnessed it, as well as to the Italian leadership.

Mussolini, who again did not appear to pay much thought to the detalis, was keen on justifying his choice to establish the puppet regime of Salò with the argument that, otherwise, the Germans would have treated Italy "ten times worse than they had treated Poland". And the fate of Poland seemed to summarize the inhumane measures taken against the populations which had fallen under German control during the eastern campaign.

As far as Mussolini was concerned, this appeared to have been enough information on the matter.

It is necessary to remember that the whole pre-war Jewish population in Italy was around 50,000, against the 500,000 German ones and many more in the Eastern territories. For this reason - if it was possible for many Germans to find good excuses not to see, it was much easier for the Italian population, and for the authorities as well, to simply disregard a problem that did not affect them directly. With this in mind, the Holocaust and its details would be treated as an "internal matter" of Germany, on which the Italians had no saying, and on which any discussion would have been pointless or dangerous of creating further friction with an ally that was necessary for the surivival of the Italian Fascist state.

I have taken these mostly from De Felice's Storia degli ebrei Italiani sotto il Fascismo and his byography of Mussolini. I would like to add that, while this should provide a more or less accurate picture of the Italian situation, it is definitly possible that there are more recent sources that I am not aware of, which may add more details or provide more evidence on those groups which participated more directly to the deportation process.