The question is to find Sylow $2$-subgroup of
1) T(Tetrahedron)
2) O(Octahedron)
3) I(Icosahedron)
From the Sylow theorem, I get the order of Sylow 2-subgroup in
T($2^2\times3$), O($2^3\times3$), I($2^2\times3\times5$) are $2^2,2^3,2^2$ respectively,and the possible $n_2$ are $(1,3)$ in T, $(1,3)$ in O, $(1,3,5,15)$ in I.
1)In a tetrahedron, there are 3 pairs of opposite edges, the line connect the centre of a pair of opposite edges generate a axis for order 2 rotation. The 3 ones and the identity forms a group of order 4($C_2\times C_2$), then there is only 1 Sylow 2-subgroup in T.
And I can not find the Sylow 2-subgroup in O and I.
Hints that may be of help:
There is an isomorphism between the symmetry group of rotations of a tetrahedron and $A_4$.
An octahedron can be inscribed within a cube, wherein a corner of the octahedron rests at the center of each face. From this, it is easy to see that every octahedron-preserving symmetry is a cube-preserving symmetry and vice-versa. This means that if you've worked with the symmetry group of a cube in the past, you might know the answer already. Otherwise, know that there is an isomorphism between the symmetry group of a cube and $S_4$.
There is an isomorphism between the symmetry group of an icosahedron and $A_5$.
Hence, we've reduced a geometric problem to a question about the symmetric and alternating groups, which could make your problem easier depending on your preferences. With some cleverness, one can even construct the isomorphism explicitly, especially in the first two parts.