If it doesn´t work, would any1 be so king to explain why to me?
Here's a direct attempt:
tutt-icosa.jpg [ 23.91 KiB | Viewed 1038 times ]
As you can see that the tips of the final icosahedron will fall out. The final result of this direct approach will be no better than lifting the pentagonal faces(the size of the pentagons will shrink meanwhile) of the tuttminx. I agree with Andreas that although there are ways to keep those tiny bits in place but needs different parts/mechanism. It is actually equivalent to some deeper cut tuttminx which we will bring up later.
As oBNoo suggested, the curvy one could be a better candidate although it is not quite a vertex-face turning icosahedron.
icosa-tutt-curvy.jpg [ 16.55 KiB | Viewed 1038 times ]