Looks like a compound of 4 icosahedra (ike) plus 2 great dodecahedra (gad), and indeed vertices coincide by ten, edges coincide by six, {6/2} coincide by pairs (type A) resp. vertices coincide by five, edges by six, {3] by four and {5} by pairs.

Incidence matrix according to Dynkin symbol

x3/2x3/2x5/2*a (type A)

.   .   .      | 120 |  1  1  1 |  1  1  1
---------------+-----+----------+---------
x   .   .      |   2 | 60  *  * |  1  1  0
.   x   .      |   2 |  * 60  * |  1  0  1
.   .   x      |   2 |  *  * 60 |  0  1  1
---------------+-----+----------+---------
x3/2x   .      |   6 |  3  3  0 | 20  *  *
x   .   x5/2*a |  10 |  5  0  5 |  * 12  *
.   x3/2x      |   6 |  0  3  3 |  *  * 20

snubbed forms: s3/2s3/2s5/2*a

s3/2s5/4s5/4*a (type B)

demi( .   .   .      ) | 60 |  2  2  2 |  1  1  1  3
-----------------------+----+----------+------------
sefa( s3/2s   .      ) |  2 | 60  *  * |  1  0  0  1
sefa( s   .   s5/4*a ) |  2 |  * 60  * |  0  1  0  1
sefa( .   s5/4s      ) |  2 |  *  * 60 |  0  0  1  1
-----------------------+----+----------+------------
      s3/2s   .        |  3 |  3  0  0 | 20  *  *  *
      s   .   s5/4*a   |  5 |  0  5  0 |  * 12  *  *
      .   s5/4s        |  5 |  0  0  5 |  *  * 12  *
sefa( s3/2s5/4s5/4*a ) |  3 |  1  1  1 |  *  *  * 60

starting figure: x3/2x5/4x5/4*a