  For the cyclic group $\mathbb{Z}_3$ and positive integer $k$, we study the
representations of the orbifold vertex operator algebra
$L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$. All the irreducible
modules for $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are classified
and constructed explicitly. Quantum dimensions and fusion rules for the
orbifold vertex operator algebra
$L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are completely determined.
