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In this article, we study the following Schrödinger equation
\begin{align*}\begin{cases}-\Delta u -\frac{\mu}{|x|^2} u+\lambda u =f(u), &\text{in}~ \mathbb{R}^N\backslash\{0\},\\\int_{\mathbb{R}^{N}}|u|^{2}\mathrm{d} x=a, & u\in H^1(\mathbb{R}^{N}),\end{cases}\end{align*}
where 
$N\geq 3$, a > 0, and 
$\mu \lt \frac{(N-2)^2}{4}$. Here 
$\frac{1}{|x|^2} $ represents the Hardy potential (or ‘inverse-square potential’), λ is a Lagrange multiplier, and the nonlinearity function f satisfies the general Sobolev critical growth condition. Our main goal is to demonstrate the existence of normalized ground state solutions for this equation when 
$0 \lt \mu \lt \frac{(N-2)^2}{4}$. We also analyse the behaviour of solutions as 
$\mu\to0^+$ and derive the existence of normalized ground state solutions for the limiting case where µ = 0. Finally, we investigate the existence of normalized solutions when µ < 0 and analyse the asymptotic behaviour of solutions as 
$\mu\to 0^-$.
