To answer this question about Japan, I'll have to explain the context. While the holocaust looms large in European history, it's not something that was particularly an issue in Japan. It's like asking what the average American thought of the Armenian genocide, back when it was discovered at the end of World War I. The average American didn't know what an Armenian was or where Armenia was--it was not a detail that fit into most people's need to know. Your average Japanese person didn't know what a Jew was, and so it was an obscure fact. Also keep in mind that most Westerners were only dimly aware of the holocaust at that time, and awareness has grown through time, but still your average person isn't too knowledgeable. There were no publicly available movies, there were no books, survivors weren't easily contactable, and there was a lot of information to assimilate about the war to figure out the scope, and what the hell had happened. I'd say most Japanese people today are only dimly aware of the Holocaust, and what they've learned mostly comes from two sources: Anne Frank and Schindler's List.

Anne Frank has enormous importance in Japan. Her Diary is very widely read and perpetually popular. Several adaptations have been made in anime and manga, plus there are stage plays, all in Japanese. But the awareness of her as Jewish, as persecuted, as a concentration camp victim, is dim. She's relatable as a child who suffered due to the war. As many many Japanese children suffered due to WWII, they empathize with that struggle. But she doesn't fit into any larger narrative of the Holocaust or Judaism. Schindler's List was popular because Spielberg is popular, but that fits into people's greater understanding as much as, say, the popularity of the movie Hotel Rwanda fits into American understanding of Rwandan politics.

FYI: there's a museum in Hiroshima dedicated to Jewish children killed in the Holocaust. Why there, why that subject? The people behind that DO understand and know about the holocaust, and their feelings are genuine. I've met them and spoken with them--they're very interested in Judaism and Jewish history, many study Jewish topics at American or Israeli universities, and many speak Hebrew. So they're an exception to the faint information most people have. I would say, though, that they do sense a lot to relate to in the story of Jews--it's largely Japanese Christians who have created this museum, and they were long persecuted in Japan, and the people of Hiroshima disproportionally have borne the suffering of WWII in Japan.

In around 1995, a magazine in Japan published an exposé--shock! It turns out that the Holocaust didn't happen and may have been made up by a larger conspiracy (according to the magazine)! Now, no serious magazine in the west would ever make the mistake of believing Holocaust deniers, and such a topic would never be even entertained as an article But the Japanese editors of the magazine were fairly clueless about the context and quality of the information they found, and genuinely felt they were on to an amazing scoop. When the context was explained to them, they apologized and retracted what they'd said, etc. The point is, the incident reveals the shallowness of knowledge about the Holocaust that is to be had in Japan, even among the well-educated. Also keep in mind that the Japanese did some pretty awful things during World War II, and had their own awful prison camps. And a lot of that information is swept under the rug in Japan. And Americans barely know about it.

So since Japan has been elaborated on, allow me to deal with Italy.

So Mussolini never really subscribed to Adolf Hitler's racial theories, saying in an interview with Emil Ludwig:

Race? It is a feeling, not a reality. Ninety-five per cent, at least. Nothing will ever make me believe that biologically pure races can be shown to exist today.… National pride has no need of the delirium of race.

And when Mussolini met Hitler for the first time in Venice, in 1934 Hitler launched into private rants about his biological theories. This bored Mussolini leading him to call Hitler a:

gramophone with just seven tunes and once he had finished playing them he started all over again

Mussolini did enact some racial laws directed at Jews in 1938, they were enacted to appease Hitler, who by that time was clearly more powerful than Mussolini.

Anyways, so Hitler and top ranking Nazis did their best to keep knowledge of the final solution from getting out to their allies; Mussolini was not told of the Wannsee conference and Hitler did not dare tell Mussolini for fear of his reaction. See the Nazis knew that the horror of the Final Solution might compel allies like Mussolini to leave the axis for fear of being associated with such an event.

Now obviously some information got out, and Italians knew vaguely that there was a large amount of violence going on in Russia and that it was directed towards Jews, but the top Italian brass didn't know the extent of the violence. Adding to that, certain Italian generals, such as the ones stationed in the Balkans saw first hand the violence that was being perpetrated by the Ustase in Croatia and that it was sanctioned by the Nazis. Furthermore the Croatians put pressure on the Italians to deport the Jews to Croatian concentration camps, but the Italians tended to deflect these requests.

The truth finally came out to the Italians in August of 1942. In a meeting between a German official named Otto Von Bismarck (Grandson of the famous Iron Chancellor I believe) and chief of staff Italian foreign minister Ciano. The Croatians had been pressuring the Italians to deport the Jews much like the Croatians were doing (the Croatian government was paid 30 marks per Jew it deported to Germany). Now the Germans began to pressure them, Bismarck let slip to the chief of staff that they were not going to be simply put to work, but that the Nazis intended for the Jews be exterminated in camps. We can't know how quickly this information worked its way up the ladder, if it did at all, but its important to note that at the very least the rumour was out there. Mussolini met Himmler in Italy and Himmler told Mussolini that Jews being deported East were being shot and worked to death, he of course added excuses that the Jews were working with the partisans.

Mussolini was now faced with incredible pressure to deal with the Jews, it was one thing to deny Croatia, but denying Nazi Germany was a much bigger issue. He immediately ordered that Italian authorities in Italian occupied Croatia were to being deporting all the Jews immediately. Why did he do this? Depending on how you look at it, Mussolini was either a pragmatist trying his best to keep up his alliance with Adolf Hitler, or he was a spineless wretch who was unable to stand up to Nazi pressure. Either way, Italy was now involved in the holocaust; but many Italians at the lower levels refused to cooperate, and the Italians were not deporting Jews from mainland Italy just yet.

Jonathan Steinberg, a historian who wrote on the holocaust from the perspective of Nazi Germany's allies has a rather amusing way of recalling the Italian resistance to the German demands that they deport their Jews.

The best tactic available was one deeply rooted in the Italian bureaucracy: to do nothing in the most officious possible way.

Italian middlemen were also aided by the fact that Mussolini's Italian empire had begun to crumble and in general Rome exercised very little direct control over these far flung occupied territories. The deportation process was slow and in other cases just flat out stalled as many Italian officials tried to desperately halt the progress of the Nazis' final solution. The Italian foreign ministry was most sympathetic to the Jews plight, and to their credit they succeeded in getting Mussolini to rescind his previous order and the Jews in Croatia, on the cusp of being deported were saved. It is a similar situation in France where the Italian authorities would do their best to obstruct the Nazis attempts to deport their Jews.

Unfortunately, it came to an end with the Italian surrender. The Germans occupied Italy and set up their own Italian puppet state. Croatia and German split Italy's former occupied territories and the final solution reached those Jews who had sough salvation in Italian territory. Mussolini was now a complete puppet of the Nazis and issued decrees declaring a virtual war on the Jews. By the end of the war, 7500 Italian Jews living in Italy were deported to camps. And the Jews living in occupied territories fared no better and were subject to the full force of the holocaust.

Now I want to clarify, I am in no way apologizing for Mussolini. The Italians did horrible things to the people living in their occupied territories. In Slovenia, they massacred huge amounts of people. And I don't want to downplay Mussolini's own treatment of the Jews. The racial legislation deprived the Jews of property, rights, and were just all around awful. But its important to recognize that even in the sea of inhumanity that was WW2, there were some people willing to take a stand.

The question arises why did the Italians oppose the holocaust so, especially when other axis members participated so willingly? Probably many reasons, but I like Steinberg's interpretation, where in he argues that opposition to the holocaust became a source of pride for the Italians who were disillusion with their German allies.

So unsurprisingly, my source recommendation is Jonathan Steinberg's "All or Nothing: The Axis and the Holocaust"