  We show the strong instability of radial ground state standing waves for the
focusing $L^2$-supercritical nonlinear Schr\"odinger equation with
inverse-square potential \[ i\partial_t u + \Delta u + c|x|^{-2} u = -
|u|^{\alpha} u, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq
3$, $u: \mathbb{R} \times \mathbb{R}^d \rightarrow \mathbb{C}$, $c\ne 0$
satisfies $c<\lambda(d):=\left(\frac{d-2}{2}\right)^2$ and $\frac{4}{d}
<\alpha<\frac{4}{d-2}$. This result extends a recent result of
Bensouilah-Dinh-Zhu [{\it On stability and instability of standing waves for
the nonlinear Schr\"odinger equation with inverse-square potential},
\url{arXiv:1805.01245}] where the stability and instability of standing waves
were shown in the $L^2$-subcritical and $L^2$-critical cases.
