$C_2$-Cofiniteness and Rationality of the Icosahedral Orbifold $V_{L_2}^{A_5}$

Let $L_2=\mathbb{Z}\alpha$ be the rank-one root lattice with $(\alpha ,\alpha )=2$, and let $A_5$ act on the lattice vertex operator algebra $V_{L_2}$ through an icosahedral subgroup of $\mathrm{Aut}(V_{L_2})\cong PS{L}_2(\mathbb{C})$. We prove that the fixed-point vertex operator algebra $V_{L_2}^{A_5}$ is $C_2$-cofinite and strongly rational.