This polyhedron can be made from 7 colors. Just like the icosahedron there will be 6 colors for your pentagon faces and same-colored pentagonal faces will be opposite from each other in the polyhedron and in parallel planes. Your triangular faces can all be the same color.
You will need to prepare 12 pentagonal faces in 6 of your colors (2 pentagons of each color) and 20 triangular faces (all of the same color).
To see the full-sized template click on this image:
To assemble this polyhedron you can surround one of the triangular faces with 3 pentagons of different colors. Around every pentagon there will be 5 triangular faces attached.
Choose one of your pentagons to think of as the central pentagon. Surround that pentagon with triangles and then attach to the surrounding triangles pentagons of the colors that have not been used in that grouping.
Surround your pentagons with triangles and then visually judge where your next pentagon shuld be attached. It, of course, will be placed on the opposite side of the polyhedron as the pentagon of the same color and will be in a parallel face to that color-matched pentagonal face.