The Kurds had very little involvement in WWII and no capacity to take action as a group. At the time, just about all of them lived in the Kurdish areas of Turkey, Iran and Iraq. They had no self-determination, and those states acted as follows:
- Turkey was neutral throughout WWII.
- Iran was invaded and occupied jointly by the USSR and the British Empire in late summer 1941, to create a route for supplies to be sent to the USSR from India, and to secure control of the Iranian oil fields. The occupying powers withdrew after the end of the war, although the Soviets did not want to at first.
- Iraq was under British influence. There was a coup by Iraqis sympathetic to Germany in spring 1941 and an attempt to expel British forces, but it failed. The British recruited local troops, about 25% of whom were Kurds, but they were under British control. As per the link:
By 1942, the Iraq Levies consisted of a Headquarters, a Depot, Specialist Assyrian companies, 40 service companies and the 1st Parachute Company, which consisted of 75% Assyrian and 25% Kurd. The new Iraq Levies Disciplinary Code was based largely on the Indian Army Act.
By 1943 the Iraq Levies strength stood at 166 British officers controlling 44 companies; 22 Assyrian, five Mixed Assyrian/Yizidi, ten Kurdish, four Marsh Arabs, and three Baluchi. Eleven Assyrian companies served in Palestine and another four served in Cyprus. The Parachute Company was attached to the Royal Marine Commando and were active in Albania, Italy and Greece. In 1943/1944 the Iraq Levies were renamed the Royal Air Force Levies.
So a platoon or so of Kurds, in the Parachute Company, served in Albania, Italy and Greece with the Royal Marine Commandos, but they weren't at Normandy. There is no sign of any Iraqi units in the order of battle for Gold, Juno or Sword beaches.
The Kurds, as a people or a political group, did not have an important role in WWII, and there was no significant Kurdish diaspora to influence events elsewhere.
Addendum: More information about the Iraq Levies here and here.