The large background triangles and pentagons that appear as you are making this polyhedron are difficult to see in this picture. You can see a large pink background triangle that seems to be facing up and to the left. You can also see a white pentagram that is facing up and tilted slightly to the front. These observations will help you place your pieces correctly and create a beautiful model.
You can make this polyhedron in 6 colors. Use 5 of the colors to make the 5-sided pieces pictured on the right of the template below and one more color to make the pentagon and the 4-sided pieces (kites) in the middle. You will need 12 pentagons of all the same color; 60 kites of that same color; and 60 5-sided pieces made from the 5 other colors.
Begin by making a cup with the pentagon at its center and the 5-sided pieces of each of your 5 colors surrounding that center pentagon (bottom right piece in the picture below). You must be consistent in how you order the colors in each cup. All of your cups in this first half of the polyhedron must be ordered in the same direction.
In his book, Polyhedron Models, Father Wenninger has an error in how he lists the colors of each cup piece. This is the corrected list. The center pentagram of each cup is the first color listed.
(1) W G R O B Y
(2) W Y B G O R
(3) W B O Y R G
(4) W O R B G Y
(5) W R G O Y B
(6) W G Y R B O
Combine 3 kites of all the same color into a dimple (bottom left piece in the picture above). You will need 5 of these dimples to surround your initial cup.
Before trying to place the second cup, you should surround your first cup with 5 of your one-colored dimples. Then you will be able to see the correct placement of your second cup.
Notice in the picture above, the yellow face of my first cup (top cup) is in the same plane as the yellow piece of my second cup (bottom left cup). You should also be able to see that the blue pieces of my first and second cups are in the same plane. Secure the second cup into its appropriate position.
The rest of your cups in this first half of the polyhedron will be placed in the same manner.
The second half of the polyheron will be composed of enantiomorphic cups (the same ordered list as above but in the reverse order). After attaching the 6th cup of the first half, you will attach the reverse of the 6th cup.