Beat the butter until light and fluffy. Gradually add the sugar, vanilla, a pinch of salt and a few drops of rum, if using. Add the eggs one by one, beating well after each addition and scraping down the sides of the bowl as necessary. Add the baking powder, ground almonds and flour and mix until well combined.
Spoon batter into the prepared springform pan and spread level.
Peel apples and cut into quarters. Cut out the core from each quarter and discard. Delete the core. Using a sharp knife, cut 4 deep slices into each quarter, cutting almost, but not all the way through. Press sliced apple quarters into the top of the batter, placing about 8 around the outside and 3 in the middle. Sprinkle sliced almonds evenly over the top.
Bake for about 60 - 70 minutes, or until the center cake tests clean with a cake tester min.
Cool the cake, the sprinkle with icing sugar to serve.