This, above, is the pentagonal icositetrahedron — one of the Catalan solids, and the dual of the snub cube. If faceting is used to remove square-based pyramids, the result is the solid shown below — a chiral polyhedron with six square faces, and 24 faces which are isosceles trapezoids.
I use a program called Stella 4d: Polyhedron Navigator (available here) to perform these polyhedral manipulations. This program has a “try to make faces regular” function, and the solid shown below is the result of using that function on the polyhedron shown above.
The “try to make faces regular” function appeared at first to leave the squares as squares (which it did), but turn the trapezoids into kites (which it did not; the yellow quadrilaterals’ longer sides are actually of two slightly different lengths). Lastly, I used Stella to create the compound of the polyhedron above and its dual. The result is shown below.