(Related question)

Let $I$ be the group of orientation preserving symmetries of a regular icosahedron. This is a $60$ element subgroup of $SO(3)$, isomorphic with the alternating group $A_5$. It is also perfect and self-normalizing in $SO(3)$.

For each $g\in SO(3)$ the conjugate ${}^gI=gIg^{-1}$ is the group of rotations which leave invariant a rotated icosahedron. My question concerns which groups can appear as intersections of conjugates. In particular, can anyone supply a proof or disproof of the following statement?

For any $g\in SO(3)$ the group ${}^gI\cap I$ is either trivial, or equal to $I$.

This elementary group theory question arose when studying numerical homotopy invariants of the Poincaré sphere $X=SO(3)/I$. In particular, I would like the fixed point sets of the two-point stabilisers of the standard action of $SO(3)$ on $X$ to be path-connected.

Thanks in advance.