Looks like a compound of 4 small stellated dodecahedra (sissid) plus 2 great icosahedra (gike), and indeed vertices coincide by ten, edges coincide by six, {10/4} coincide by pairs.

Incidence matrix according to Dynkin symbol

x5/4x5/4x3/2*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
---------------+-----+----------+---------
x5/4x   .      |  10 |  5  5  0 | 12  *  *
x   .   x3/2*a |   6 |  3  0  3 |  * 20  *
.   x5/4x      |  10 |  0  5  5 |  *  * 12

snubbed forms: s5/4s5/4s3/2*a