Denote the space of functions on the set of faces of the icosahedron by $V_f$, this is a 20-dimensional representation of $A_5$ which acts here on as the group of rotational symmetries. Here follows my attempt (which seems to fail):
The character table of $A_5$, which will be needed:
There is a 9 dimensional subrepresentation obtained as follows. There is a 10-dimensional sub representation, call this $\mathbb{C}^{10}$, by assigning values to pairs of opposite faces of the icosahedron. Then with the condition that summing of all these values will give zero this sub representation of the sub representation $\mathbb{C}^{10}$ has dimension 9, which I will denote by $\mathbb{C}^9$. Now I want to decompose this representation using the character table of $A_5$, since $\chi_{\mathbb{C^{10}}}(g)=\chi_{\mathbb{C^{1}}}(g)+\chi_{\mathbb{C^{9}}}(g)=1+\chi_{\mathbb{C^{9}}}(g)$.
For $g=(123)$, which the rotation of order three on the axis through opposite faces, we see that only one pair of faces is fixed, so $\chi_{\mathbb{C^{10}}}((123))=1$, so we see that $\chi_{\mathbb{C^{9}}}((123))=0$. This can be obtained if it were the case that $\mathbb{C}^9=\mathbb{C}^4\oplus\mathbb{C}^5$. Of course we must try some more elements to guarantee that this is the case.
For $g=(12345)$, which is the rotation of order 5 obtained by rotating on the axis through a pair of vertices, we have that no pair of faces is fixed, so $\chi_{\mathbb{C^{10}}}((12345))=0$, which implies $\chi_{\mathbb{C^{9}}}((12345))=-1$ which again is obtained if $\mathbb{C}^9=\mathbb{C}^4\oplus\mathbb{C}^5$.
Now if $g=(12)(34)$ I get problems. This element is obtained by rotation of order 2 on the axis through opposite edges. This does also not fix any pair of faces, so $\chi_{\mathbb{C^{10}}}((12)(34))=0$ which implies again $\chi_{\mathbb{C^{9}}}((12)(34))=-1$, but $\chi_{\mathbb{C^{4}}}((12)(34))+\chi_{\mathbb{C^{5}}}((12)(34))=1$ so it cannot be the case that $\mathbb{C}^9=\mathbb{C}^4\oplus\mathbb{C}^5$.
I dont understand what is going wrong here, because using the other irreducible representations I cannot make it consistent with the values of the characters. I would really appreciate if someone could clarify where I am mistaken. Thank you.
(12)(34) stabilizes two pairs of opposite faces (because, in each case, it swaps them).