This was a splinter group from the Catholic Church which was devoted to bringing about the horrors of the Mythos upon humanity, so that they would have reason to return to God. To further their goals, they sought out a mysterious work called the de'Medici Manoscritto, a work which gave non-Sanity-draining equivalents to Mythos spells.
Given the fact the DG source book has named Hitler, Himmler and the Nazis as mythos worshipping occultists from Hell, there is now no reason not to ascribe a Mythos rationale for their occurrence to other unpleasant events of WW2, as and when required by keepers for dramatic purposes.
The one I have in mind is the supposed acquiescence (and becoming therefore an accomplice of the fascists) of the Vatican in the persecution of the Jews. The commonly accepted reason is anti-Semitism within the Church of the period, as well as fear of retribution. This is probably true in the real world. At this point it should be remembered that many individual members of the Church did help minorities persecuted by the fascists during WW2.
However in the DG world, what if the Last Dawn had survived the 1920s/1930s into the 1940s?
The Last Dawn should have seen the Nazis coming before they came to power as statements made by several prominent Nazis were definitely unchristian. They may well have aided their path to power, using their world-wide chapter structure to influence people into supporting the Nazis. A study of their rise to power seems to show that more than a modicum of very good luck, excellent networking skills and the assistance of people with other causes helped the Nazis into government. The actions of the Nazis in power should have removed any Last Dawn doubts about what they believed.
By actively assisting the Nazis, the Last Dawn would be helping to fulfil it's own mission. Prejudice would probably come into play into the game world as well; the rationale being that it would be good that Jews and heathens (fascists) were helping to bring about the end of the world, and not 'good Christians'.
Having the Last Dawn in the driving seat in Vatican relations with fascists would make for an interesting environment to play in. In a Resistance/Maquis/Jedburg team based WW2 DG game, the Church could no longer be relied upon as a refuge, and any contacts or services supplied by priests would have to be tested by P4/Pisces operatives before being relied on. This could be exceptionally potent, if used during 1945 as the remnants of the Karotechia are hunted down.
Based on material submitted to the DGML by Daniel Harms and Adam Crossingham.