Charcoal, 162 146 bytes
Ｎθ⊞υ⟦⁰¦¹⊘⊕₂⁵⟧≔⟦⟧ηＦυＦＥ²⊞ＯΦιν×⊖⊗κ§ι⁰Ｆ¬№υκＦΦ⊞Ｏυκ⁼⁴ΣＸＥμ⁻ξ§κπ²⊞η⟦κμ⟧ＦηＦ…¹θ⊞υＥ§ι⁰⁺×κλ×⁻θꧧι¹μＦηＦΦυ⬤ι№η⟦μκ⟧Ｆ…²θＦ…¹λ⊞υＥ§ι⁰⁺⁺×μν×⁻λ짧ι¹ξ×⁻θλ§κξＩＥυ∕ι₂ΣＸι² Try it online! Link is to verbose version of code. Explanation:
Ｎθ Input b.
Ｎθ⊞υ⟦⁰¦¹⊘⊕₂⁵⟧≔⟦⟧ηＦυＦＥ²⊞ＯΦιν×⊖⊗κ§ι⁰Ｆ¬№υκＦΦ⊞Ｏυκ⁼⁴ΣＸＥμ⁻ξ§κπ²⊞η⟦κμ⟧⊞υ⟦⁰¦¹⊘⊕₂⁵⟧≔⟦⟧η Generate the verticesStart with one vertex and no edges of a regularan icosahedron of side 2.
ＦηＦ…¹θ⊞υＥ§ι⁰⁺×κλ×⁻θꧧι¹μＦυ ExtrapolateProcess all vertices as they are discovered.
ＦＥ²⊞ＯΦιν×⊖⊗κ§ι⁰ Generate more vertices by rotating about the line x=y=z and reflecting in the xy plane. (This is the same procedure as for the regular dodecahedron in the linked question.)
Ｆ¬№υκ Ignore previously discovered vertices.
ＦΦ⊞Ｏυκ⁼⁴ΣＸＥμ⁻ξ§κπ²⊞η⟦κμ⟧ Save the new vertex and also discover any edges by checking for previously found vertices that are 2 away. (Very fortunately, ϕ²+1²+(ϕ-1)²=2² under floating-point arithmetic.) Since only previously found vertices are checked, each edge is only detected once, and has a specific direction.
Ｆη Loop over the edges.
Ｆ…¹θ Loop over the intermediate points.
⊞υＥ§ι⁰⁺×κλ×⁻θꧧι¹μ Extrapolate a point along the edge. (Sadly Charcoal doesn't have a way of adding two vectors which would really simplify this. Maybe if it had octonions...)
Ｆη Loop over the edges again.
ＦηＦΦυ⬤ι№η⟦μκ⟧Ｆ…²θＦ…¹λ⊞υＥ§ι⁰⁺⁺×μν×⁻λ짧ι¹ξ×⁻θλ§κξＦΦυ⬤ι№η⟦μκ⟧ ExtrapolateLoop over the vertices that have edges to both vertices of that edge. (Note that because the edges have a specific direction, each face can only be detected once.)
Ｆ…²θＦ…¹λ Loop over the interior points of the face.
⊞υＥ§ι⁰⁺⁺×μν×⁻λ짧ι¹ξ×⁻θλ§κξ Extrapolate a point inside the facesface. (Due to the normalisation, it's not necessary to interpolate.)
ＩＥυ∕ι₂ΣＸι² Normalise all the points.