I hope one of our trusted contributors on Finland will show up and go into further detail, especially concerning the debates in Finland this discovery caused but as far as the history of the EK Finland goes, here's what Oula Silvennoinen, the PhD student, discovered:
Silvennoinen shows that the Finnish secret police Valtiollinen Poliisi (Valpo) worked in close cooperation with the German Reich Security Main Office and its predecessor organizations. While he describes several phases of cooperation, in general, the two organizations worked closely together. This was due not only through personal connections such as those of Wenrer Best and Heinrich Müller of Gestapo fame and the second in command of Valpo, Bruno Aaltonen, but also due to ideological affinities. What unite Valpo and the Nazis was anti-Communism.
The anti-communist bend of Valpo contributed according to Silvennoinen majorly to the close cooperation of Valpo and the RSHA, especially in the interrogation, selection, and execution of Soviet POWs in Finnish POW camps. Silvennoinen explains that Finland and with it, Valpo, had due to their own experience first with attempted communist revolution after WWI and after that with Soviet designs on expansion and the declaration of the Winter War, slid more and more towards, what he terms "anti-Communist democracy". Valpo was especially affected due to its institutional nature as a secret police engaged in fighting communist and agitators. Due to this ideological affinity Valpo also started more and more to warm towards Nazi anti-Semitism, which saw Jews as responsible for Bolshevism and thus participate in anti-Semitic campaigns of the Nazis.
Silvennoinen's main point concerning Valpo is that at some point the institution started to operate, especially in its collaboration with the RSHA, with only minimal or no knowledge of the democratic government as far as can be told, though he makes the point that further studies would be necessary to illuminate this particular point. Over all however, he draws a picture of a democratic state collborating with the Nazis not because of occupation but because of geopolitical and security political considerations.
Concerning the EK Finland, he shows that the unit, founded in Summer 1941 and dissolved in autumn 1942, did indeed participate in Nazi crimes by selecting up to 3000 people who had been captured by Finnish forces during the continuation war and kept in Finnish POW camps top be shot by the Nazis. He not only shows that "Jew" was a specific category for selection but also that the EK Finland did in fact participate in anti-Partisan operations of the Germans, meaning that they were involved in the killing of civilians and Jews, in the Northern Soviet Union.
The project during the course of which Silvennoinen researched his dissertation shows that although Finnish courts did indeed sentence people responsible for the unlawful killing of Soviet POWs, EK Finland did not play a role during these proceedings, which deaths with only 56 cases of unlawful deaths of POWs – a number he terms to be too small.
Silvennoinen's contribution to this volume is especially fascinating since it deals with another of Valpo's tasks: Handing over civilians to the German authorities for either deportation or killing, including political adversaries of Nazi Germany who had fled to Finland. Though this concerned "only" 135 cases it still is a chapter in Finish history that has been neglected so far and where one of the most important factors was that this happened in accordance with existing legislation.
Basically, this gives a quick summary of Silvennoinen's points as far as I am familiar with them. His thesis has been published in German but not in English as far as I am aware. I really hope someone might illuminate further on the discussions that arose from his findings within Finnish public discourse.
Edit: I also highly recommend
this article going into further detail. It' title is Antero Holmila: Finland and the Holocaust: A Reassessment. from: Holocaust and Genocide Studies.