The augmented icosahedron is is made by attaching one (tetrahedron) pyramid to each face of the [simple icosahedron|icosahedron#simple], retaining the same symmetries.

images/constructed/dscn1651.jpg		images/constructed/dscn1652.jpg		images/constructed/dscn1653.jpg

In total you'll need:

COMPONENTS			AMOUNT		NOTES
vertices of 3		20			pyramid pinnacles, one per face of icosahedron
vertices of 10		12			icosahedron vertices
edges 				90			30 (icosahedron) + 20 x 3 (pyramid extras)	

images/constructed/dscn1658.jpg
((An insider look at the different types of vertices: bright pink (3-vertices), dark purple (10-vertices)))

The augmented icosahedron is part of the [icosahedron-in-dodecahedron].