  An irreducible module for the parafermion vertex operator algebra
$K(\mathfrak{sl}_2,k)$ is said to be of $\sigma$-type if an automorphism of the
fusion algebra of $K(\mathfrak{sl}_2,k)$ of order $k$ is trivial on it. For any
integer $k \ge 3$, we show that there exists an automorphism of order $2$ of
the subalgebra of the fusion algebra of $K(\mathfrak{sl}_2,k)^{\langle \theta
\rangle}$ spanned by the irreducible direct summands of $\sigma$-type
irreducible $K(\mathfrak{sl}_2,k)$-modules, where $\theta$ is an involution of
$K(\mathfrak{sl}_2,k)$. We discuss some examples of such an automorphism as
well.
