Looks like a compound of 2 small stellated dodecahedra (sissid), and indeed edges and faces both coincide by pairs, but vertices are identified.

Incidence matrix according to Dynkin symbol

x5/2o5/2o5/2*a

.   .   .      | 12 | 10 |  5  5
---------------+----+----+------
x   .   .      |  2 | 60 |  1  1
---------------+----+----+------
x5/2o   .      |  5 |  5 | 12  *
x   .   o5/2*a |  5 |  5 |  * 12

x5/3o5/3o5/2*a

.   .   .      | 12 | 10 |  5  5
---------------+----+----+------
x   .   .      |  2 | 60 |  1  1
---------------+----+----+------
x5/3o   .      |  5 |  5 | 12  *
x   .   o5/2*a |  5 |  5 |  * 12

o5/3x5/3o5/2*a

.   .   . | 12 | 10 |  5  5
----------+----+----+------
.   x   . |  2 | 60 |  1  1
----------+----+----+------
o5/3x   . |  5 |  5 | 12  *
.   x5/3o |  5 |  5 |  * 12

o5/2o5β

both( .   . . ) | 12 | 10 |  5  5
----------------+----+----+------
sefa( .   o5β ) |  2 | 60 |  1  1
----------------+----+----+------
      .   o5β   ♦  5 |  5 | 12  *
sefa( o5/2o5β ) |  5 |  5 |  * 12

starting figure: o5/2o5x