Here is my work so far:
Let $G$ denote the group of orientation-preserving isometries of Icosahedron. I have shown that $G$ acts transitively on $G/G_s$ where $G_s$ is the isotropy group, namely $G_s=\{g \in G:g(s)=s\}$ where $s$ could be a vertex, an edge or a face in the Icosahedron. Furthermore, I have shown that this implies that for any normal subgroup $N\subseteq G $, following holds: $$g(Nx)=Ng(x),\quad \forall \ x\in G/G_s, \ \forall g\in G,$$ where $Nx=\{n(x):n\in N\}$ is the orbit of $x$.
I know that $\{g\in G:\mbox{order}(g)=5\}$ can only be the group of rotations around a vertex by angles of $\frac{2\pi}{5},\frac{4\pi}{5},\frac{6\pi}{5},\frac{8\pi}{5},\mbox{ and }2\pi.$ I am planning to show that this group cannot be normal by showing the above equation breaks down. However, I cannot find exlicit example of $x\in G/G_s$.
Any help will be highly appreciated.
Using Lagrange's theorem, we know that $|N|=5,|G|=60$ implies that $|G/N|=\frac{60}{5}=12=|V|$. The action of $G$ on $V$ is transitive. Hence $$\{g\in G:\mbox{order}(g)=5\}$$ is the group of rotations around a vertex $v\in V$ by angles of $\frac{2\pi}{5},\frac{4\pi}{5},\frac{6\pi}{5},\frac{8\pi}{5},2\pi.$ Let this group $G_v:=\{g\in G:g(v)=v\}$ be denoted by $N$. We show $N$ is not normal in $G.$ It is enough to find $g\in G$ such that the normality condition breaks down, i.e. $$g^{-1}Ng\neq N.$$ Let $g\in G$ be a rotation around the center of a face whose one vertex is $v\in V$ by angle of $\frac{2\pi}{3}$. Let $v_1,v_2\in V$ be the other two vertices of the face. Then, $g^{-1}ng(v)=g^{-1}v(v_1)\neq v,$ if $n(v_1)\neq v_1, \ n\in N.$ But $n(v)=v, \ \forall n\in N.$ It follows that $g^{-1}ng \notin N,$ if $n\neq id_G.$ Thus, $g^{-1}Ng\neq N.\Rightarrow N$ is not normal in $G.$