The net which generates an icosahedron, will also generate a number of other polyhedra. Each of these polyhedra is 'isomorphous' to the icosahedron. They each consist of twenty triangular faces, five of which are connected to each one of twelve vertices.
I recently attempted to generate and catalogue the isomorphs to the icosahedron. To my surprise there were a number of additional isomorphs in addition to the ones that I predicted. To my knowledge the total number of isomorphs is unknown.
'SemiRegular' Isomorphs
The term 'semiregular' is my own term for these and reflects the fact that they are derived from a regular polyhedron. The names of the semiregular isomorphs are also of my own invention.
The most obvious isomorph
to the icosahedron can be obtained by inverting one of the vertices of
the regular icosahedron. I term this an inverted
icosahedron. Any other nonneighbouring vertex can also be inverted,
if this is opposite the first vertex I term the resultant polyhedron a
parabiinverted
icosahedron. The inversion of one of the five vertices surrounding
this opposite vertex results in a metabiinverted
icosahedron. From this polyhedron a third vertex can be inverted
to generate a triinverted icosahedron. These four
polyhedra are shown below.
Inverted icosahedron 
Metabiinverted icosahedron 
Parabiinverted icosahedron 
Triinverted icosahedron 
The great
icosahedron (above) is also isomorphous to the Icosahedron. A similar
process to that above can be performed. In these cases I term the adjusted
vertices as 'everted'. The result of this eversion is a pentagrammic
pyramid pointing out from the remainder of the polyhedron. As above, four
isomorphs can be produced: the everted great icosahedron,
the parabieverted great icosahedron, the metabieverted
great icosahedron and the trieverted great icosahedron.
These four polyhedra are shown below.
everted great icosahedron 
metabieverted great icosahedron 
parabieverted great icosahedron 
trieverted great icosahedron 
Pentagrammic AntiPrism
Three further isomorphs can
be produced starting from the base of a pentagrammic
antiprism. Replace
the two pentagrammic faces with pentagrammic pyramids pointing out from
the antiprism to generate a bieverted pentagrammic
antiprism. With both pyramids pointing inwards a a biinverted
pentagrammic antiprism results. One pyramid can point each way
to form an evertedinverted pentagrammic antiprism.
These three polyhedra are shown below.
bieverted pentagrammic antiprism 

evertedinverted pentagrammic antiprism 
Triangular Snub Antiprisms
3/2 snub antiprism 
great 3/2 snub antiprism 
The icosahedron is also the triangular member of an infinite family of prismatic 2uniform polyhedra, the snub antiprisms. As such, the various isomorphs to the snub antiprisms can also be found in the isomorphs to the icosahedron. Above are two such examples, the 3/2 snub antiprism and the great 3/2 snub antiprism. The 33/2 hybrid snub antiprism and the inverted triangular snub antiprism also exist but we have already come across these in the guise of the triinverted icosahedron and the trieverted great icosahedron respectively.
Other isomorphs
When I attempted to generate the isomorphs of the icosahedron, I imagined that the above polyhedra would be the complete set (I will admit that the snub antiprisms were not in my original list). To my surprise a large number of other isomorphs were produced. One single run of HEDRON produced over eighty of these (discounting degenerate cases with superposed vertices or faces). They are difficult to categorise as they show a variety of starred, crossed and uncrossed vertices, and also show various degrees of symmetry. Some of the more interesting of these isomorphs are shown below, many are similar and some appear to be no more than random aggregations of triangles. Any suggestions on their categorisation or on the ennumeration of the total number of possible isomorphs would be welcomed.
One 'hint' as to the form of these 'other' isomorphs is that they appear to need each vertex to be nonregular. This supposition is made as an attempt to generate isomorphs with one vertex held regular by the insertion of a pentagonal or pentagrammic 'brace' inside the model resulted in only the 'semiregular' isomorphs being formed.
With regard to the examples
below, note the relationship betweens examples 1
and 5 and , the similarity between examples 3
and 4.
Example 1 
Example 2 
Example 3 
Example 4 
Example 5 
Example 6 
Example 7 
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