From what i know, they supported Finland during the war, sending supplies and volunteers to fight for the Fins. They took inn a great deal of both Finnish, Danish, Norwegian and Jewish refugees during the war.
Sold a good amount of steel to the German war machine and helped train Norwegian police units while the Norwegian government were in exile. later on they supported the Norwegian and Soviet campaign in Finnmark (44-45).
They also sent state officials to Vinnitsa and Katyn in 1943 during the German led investigation of the massacre that took place there during the Soviet purges of the 1930's, witch became a huge deal back home and in most German occupied territories.
The Swedish Red Cross also undertook the White buses expedition late in the war, in an effort to rescue Scandinavian prisoners in the many concentration camps under Nazi control.
Despite all of this they remained largely neutral during the war, but feared both Nazi and allied invasion during the war.