  In this paper, we study the scattering theory for the cubic inhomogeneous
Schr\"odinger equations with inverse square potential $iu_t+\Delta
u-\frac{a}{|x|^2}u=\lambda |x|^{-b}|u|^2u$ with $a>-\frac14$ and $0<b<1$ in
dimension three. In the defocusing case (i.e. $\lambda=1$), we establish the
global well-posedness and scattering for any initial data in the energy space
$H^1_a(\mathbb R^3)$. While for the focusing case(i.e. $\lambda=-1$), we obtain
the scattering for the initial data below the threshold of the ground state, by
making use of the virial/Morawetz argument as in Dodson-Murphy [Proc. Amer.
Math. Soc.,145(2017), 4859-4867.] and Campos-Cardoso [arXiv: 2101.08770v1.]
that avoids the use of interaction Morawetz estimate.
