Augmenting the icosahedron
The icosahedron, vertex figure (3,3,3,3,3)
Left: Each triangular face of the icosahedron can be excavated with a triangular cupola completing a cycle around the axis. The triangular caps of the cupola are coincident to the original triangular faces. The 3's in the vertex figure are each replaced by (4,3,4) The original vertex figure becomes (4,3,4,4,3,4,4,3,4,4,3,4,4,3,4)/4 or ((4,3,4)5)/4 with new vertices of (6,4,3). All vertices lie on the convex hull.
Centre: The left-hand polyhedron can be further augmented by augmenting the hexagonal faces of the triangular cupolas with a further triangular cupola, such that the square faces of the two sets of cupolas are edge connected. This is equivalent to excavating a triangular orthobicupola from the original icosahedron. The original vertex figure of (4,3,4,4,3,4,4,3,4,4,3,4,4,3,4)/4 or ((4,3,4)5)/4 is unchanged. The vertices of (6,4,3) become (3,4,4,3). New vertices are added of the form (3,4,3,4). The square and triangular faces of the new cupolas are shown in red and yellow respectively. All vertices lie on the convex hull.
Right: Another augmentation of the icosahedron has each face of the icosahedron excavated with a triangular pyramid or tetrahedron, each 3 in the vertex figure is replaced by a (3,3) completing a cycle around the axis. The original vertex figures become (3,3,3,3,3,3,3,3,3,3)/3 - or (310)/3 with new vertices of (3,3,3). All vertices are on the exterior of the figure. This figure nearly escaped me as the (3,3,3) vertices are only barely visible. To see these vertices click on the linked image above and zoom into one of the triangular valleys.