  In this paper, we study the stability and instability of the ground states
for the focusing inhomogeneous nonlinear Schr\"{o}dinger equation with
inverse-square potential (for short, INLS$_c$ equation): \[iu_{t} +\Delta
u+c|x|^{-2}u+|x|^{-b} |u|^{\sigma } u=0,\; u(0)=u_{0}(x) \in H^{1},\;(t,x)\in
\mathbb R\times\mathbb R^{d},\] where $d\ge3$, $0<b<2$,
$0<\sigma<\frac{4-2b}{d-2}$ and $c\neq 0$ be such that
$c<c(d):=\left(\frac{d-2}{2}\right)^{2}$. In the mass-subcritical case
$0<\sigma<\frac{4-2b}{d}$, we prove the stability of the set of ground states
for the INLS$_{c}$ equation. In the mass-critical case $\sigma=\frac{4-2b}{d}$,
we first prove that the solution of the INLS$_c$ equation with initial data
$u_{0}$ satisfying $E(u_0)<0$ blows up in finite or infinite time. Using this
fact, we then prove that the ground state standing waves are unstable by
blow-up. In the intercritical case $\frac{4-2b}{d}<\sigma<\frac{4-2b}{d-2}$, we
finally show the instability of ground state standing waves for the INLS$_c$
equation.
