Looks like a compound of 2 rhombidodecadodecahedron (raded), and indeed vertices, edges, and squares (type A), plus pentagons (type B) respective pentagrams (type C) all coincide by pairs.

Incidence matrix according to Dynkin symbol

x5/2β5x   (type A)

both( .   . . ) | 120 |  1  1  1  1 |  1  1  1  1
----------------+-----+-------------+------------
both( x   . . ) |   2 | 60  *  *  * |  1  1  0  0
both( .   . x ) |   2 |  * 60  *  * |  1  0  1  0
sefa( x5/2β . ) |   2 |  *  * 60  * |  0  1  0  1
sefa( .   β5x ) |   2 |  *  *  * 60 |  0  0  1  1
----------------+-----+-------------+------------
both( x   . x ) |   4 |  2  2  0  0 | 30  *  *  *
      x5/2β .   ♦  10 |  5  0  5  0 |  * 12  *  *
      .   β5x   ♦  10 |  0  5  0  5 |  *  * 12  *
sefa( x5/2β5x ) |   4 |  0  0  2  2 |  *  *  * 30

starting figure: x5/2x5x

β5/2β5x   (type C)

both( .   . . ) | 120 |  1   2  1 |  1  1  2
----------------+-----+-----------+---------
both( .   . x ) |   2 | 60   *  * |  0  1  1
sefa( s5/2s . ) |   2 |  * 120  * |  1  0  1
sefa( .   β5x ) |   2 |  *   * 60 |  0  1  1
----------------+-----+-----------+---------
both( s5/2s . ) ♦   5 |  0   5  0 | 24  *  *
      .   β5x   ♦  10 |  5   0  5 |  * 12  *
sefa( β5/2β5x ) |   4 |  1   2  1 |  *  * 60

starting figure: x5/2x5x

x5/2β5β   (type B)

both( .   . . ) | 120 |  1   2  1 |  1  1  2
----------------+-----+-----------+---------
both( x   . . ) |   2 | 60   *  * |  0  1  1
sefa( .   s5s ) |   2 |  * 120  * |  1  0  1
sefa( x5/2β . ) |   2 |  *   * 60 |  0  1  1
----------------+-----+-----------+---------
both( .   s5s ) ♦   5 |  0   5  0 | 24  *  *
      x5/2β .   ♦  10 |  5   0  5 |  * 12  *
sefa( x5/2β5β ) |   4 |  1   2  1 |  *  * 60

starting figure: x5/2x5x