Looks like a compound of a small rhombidodecahedron (sird) plus 12 {10/2} fitting into the dodecahedral set of holes, and indeed vertices and {10/2}-edges both coincide by pairs.

Incidence matrix according to Dynkin symbol

x5/2x5x

.   . . | 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
--------+-----+----------+---------
x5/2x . |  10 |  5  5  0 | 12  *  *
x   . x |   4 |  2  0  2 |  * 30  *
.   x5x |  10 |  0  5  5 |  *  * 12

snubbed forms: β5/2x5x, x5/2β5x, x5/2x5β, β5/2β5x, β5/2x5β, x5/2β5β, s5/2s5s