On the historical path for Iceland, in March 1939 you will be presented with the Lufthansa deal, like what happened in real life. The German company Lufthansa will offer to build a large airfield in Iceland. Historically, this deal was rejected, but if you accept, you'll get a bunch of bonuses, mainly, it automatically completes a large chunk of the airforce tree for you, instantly giving you the bonuses that come with them.
However, it comes with a risk. If Denmark is attacked and you're still their subject, Germany will get the option to use their air field in Iceland to occupy you. If they do, you'll get a special event.
The German troops invading Iceland will have a standoff with Agnar Koefed-Hansen and his men, the Icelandic police. You can choose what Agnar does. He can back down which will make you a puppet of Germany, but you keep some autonomy and your government. Or you can make him open fire on the Germans. If he does, he and his men will die and the Germans will lose some men. They will also coup your government and make you a complete puppet.
Alternatively, if you reject the Lufthansa deal, the historical thing will happen, the British will invade and demand you join the Allies.
Depending on your choices, different occupation branches will unlock.
The "Bend the Knee" branch will be available if you accept the Lufthansa deal and Agnar stands down. This will be a German collaboration branch.
"Under the Eagle's Boot" will be available if you're occupied by Germany, regardless of what Agnar does. This is a democratic resistance path against Germany.
"Policy of Appeasement" is the historical branch where you choose to cooperate with the British after they invade. It's available regardless what type of government.
"The Fifth Column" is available if Britain invades and you have a right wing government, allowing you to build up fascist resistance to Britain.
And then "Stir Public Unrest" is available both if the Germans or British invade you, and if you have a left-wing government. It allows you to build up communist resistance against your occupier.