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We consider the $L^2$-critical NLS with inverse-square potential  $$  i \partial_t u +\Delta u + c|x|^{-2} u = -|u|^{\frac{4}{d}} u, \quad u(0) =u_0, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d,  $$  where $d\geq 3$ and $c\ne 0$ satisfies $c<\lambda(d) :=\left(\frac{d-2}{2}\right)^2$. Using a refined compactness lemma, we extend themass concentration of finite time blow-up solutions established in theattractive case by the first author in [Bensouilah] to $c<\lambda(d)$. By meansof a simple and short limiting profile theorem, we get the same classificationresult obtained by Csobo and Genoud in [CsoboGenoud] for $0<c<\lambda(d)$. Italso enables us to extend the classification to $c<\lambda(d)$.

Mass concentration and characterization of finite time blow-up solutions for the nonlinear Schr\"odinger equation with inverse-square potential