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⟧

Generate the vertices and edges of a regular icosahedron of side 2.

ＦηＦ…¹θ⊞υＥ§ι⁰⁺×κλ×⁻θꧧι¹μ

Extrapolate the points along the edges.

ＦηＦΦυ⬤ι№η⟦μκ⟧Ｆ…²θＦ…¹λ⊞υＥ§ι⁰⁺⁺×μν×⁻λ짧ι¹ξ×⁻θλ§κξ

Extrapolate the points inside the faces. (Due to the normalisation, it's not necessary to interpolate.)

ＩＥυ∕ι₂ΣＸι²

Normalise all the points.

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.

⊞υ⟦⁰¦¹⊘⊕₂⁵⟧≔⟦⟧η

Start with one vertex and no edges of an icosahedron of side 2.

Ｆυ

Process 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.

ＦΦυ⬤ι№η⟦μκ⟧

Loop 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 face. (Due to the normalisation, it's not necessary to interpolate.)

ＩＥυ∕ι₂ΣＸι²

Normalise all the points.

Charcoal, 162 bytes

Ｎθ⊞υ⟦⁰¦¹⊘⊕₂⁵⟧ＦυＦＥ²⊞ＯΦιν×⊖⊗κ§ι⁰Ｆ¬№υκ⊞υκＦ¹²«≔§υιηＦι«≔§υκζ¿⁼⁴ΣＸＥη⁻λ§ζ첫Ｆ…¹θ⊞υＥη⁺×λμ×⁻θλ§ζνＦ…υκ¿⁼⁴ΣＸＥη⁻μ§λν²¿⁼⁴ΣＸＥζ⁻μ§λν²Ｆ…²θＦ…¹μ⊞υＥη⁺⁺×νξ×⁻μν§ζπ×⁻θμ§λπ»»»ＩＥυ∕ι₂ΣＸι²

Try it online! Link is to verbose version of code. Explanation:

Ｎθ

Input b.

⊞υ⟦⁰¦¹⊘⊕₂⁵⟧ＦυＦＥ²⊞ＯΦιν×⊖⊗κ§ι⁰Ｆ¬№υκ⊞υκ

Generate the vertices of a regular icosahedron of side 2.

Ｆ¹²«≔§υιηＦι«≔§υκζ¿⁼⁴ΣＸＥη⁻λ§ζ첫Ｆ…¹θ⊞υＥη⁺×λμ×⁻θλ§ζν

Extrapolate between each pair of vertices that are distance 2 apart, plus...

Ｆ…υκ¿⁼⁴ΣＸＥη⁻μ§λν²¿⁼⁴ΣＸＥζ⁻μ§λν²Ｆ…²θＦ…¹μ⊞υＥη⁺⁺×νξ×⁻μν§ζπ×⁻θμ§λπ»»»

... each triplet of vertices that are pairwise distance 2 apart. (Due to the normalisation, it's not necessary to interpolate.)

ＩＥυ∕ι₂ΣＸι²

Normalise all the points.

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⟧

Generate the vertices and edges of a regular icosahedron of side 2.

ＦηＦ…¹θ⊞υＥ§ι⁰⁺×κλ×⁻θꧧι¹μ

Extrapolate the points along the edges.

ＦηＦΦυ⬤ι№η⟦μκ⟧Ｆ…²θＦ…¹λ⊞υＥ§ι⁰⁺⁺×μν×⁻λ짧ι¹ξ×⁻θλ§κξ

Extrapolate the points inside the faces. (Due to the normalisation, it's not necessary to interpolate.)

ＩＥυ∕ι₂ΣＸι²

Normalise all the points.