  In this paper, we study the Cauchy problem for the energy-critical
inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential
\[iu_{t} +\Delta u-c|x|^{-2}u=\lambda|x|^{-b} |u|^{\sigma } u,\; u(0)=u_{0} \in
H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $\lambda=\pm1$,
$0<b<2$, $\sigma=\frac{4-2b}{d-2}$ and
$c>-c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first prove the local
well-posedness as well as small data global well-posedness and scattering in
$H^{1}$ for $c>-\frac{(d+2-2b)^{2}-4}{(d+2-2b)^{2}}c(d)$ and $0<b<\frac{4}{d}$,
by using the contraction mapping principle based on the Strichartz estimates.
Based on the local well-posedness result, we then establish the blowup criteria
for solutions to the equation in the focusing case $\lambda=-1$. To this end,
we derive the sharp Hardy-Sobolev inequality and virial estimates related to
this equation.
