Looks like a compound of 2 great icosahedra (gike), and indeed edges and faces both coincide by pairs, but vertices are identified.

As abstract polytope 2gike is isomorphic to 2ike.

Incidence matrix according to Dynkin symbol

o3/2x3o5*a

.   . . | 12 | 10 |  5  5
--------+----+----+------
.   x . |  2 | 60 |  1  1
--------+----+----+------
o3/2x . |  3 |  3 | 20  *
.   x3o |  3 |  3 |  * 20

o5/4o3x3*a

.   . .    | 12 | 10 |  5  5
-----------+----+----+------
.   . x    |  2 | 60 |  1  1
-----------+----+----+------
.   o3x    |  3 |  3 | 20  *
o   . x3*a |  3 |  3 |  * 20

o5/4o3/2x3/2*a

.   .   .      | 12 | 10 |  5  5
---------------+----+----+------
.   .   x      |  2 | 60 |  1  1
---------------+----+----+------
.   o3/2x      |  3 |  3 | 20  *
o   .   x3/2*a |  3 |  3 |  * 20