  We consider the inhomogeneous nonlinear Schr\"odinger equation with
inverse-square potential in $\mathbb{R}^N$ $$ i u_t + \mathcal{L}_a u+\lambda
|x|^{-b}|u|^\alpha u = 0,\;\;\mathcal{L}_a=\Delta -\frac{a}{|x|^2}, $$ where
$\lambda=\pm1$, $\alpha,b>0$ and $a>-\frac{(N-2)^2}{4}$. We first establish
sufficient conditions for global existence and blow-up in $H^1_a(\mathbb{R}^N)$
for $\lambda=1$, using a Gagliardo-Nirenberg-type estimate. In the sequel, we
study local and global well-posedness in $H^1_a(\mathbb{R}^N)$ in the
$H^1$-subcritical case, applying the standard Strichartz estimates combined
with the fixed point argument. The key to do that is to establish good
estimates on the nonlinearity. Making use of these estimates, we also show a
scattering criterion and construct a wave operator in $H^1_a(\mathbb{R}^N)$,
for the mass-supercritical and energy-subcritical case.
