Summary: In this paper, we look for normalized solutions to the following non-autonomous Schrödinger equation \[ \begin{cases} -\Delta u=\lambda u+h(x)|u|^{q-2}u+|u|^{2^*-2}u \text{ in } \mathbb{R}^N, \\ \int_{\mathbb{R}^N} |u|^2 \mathrm{d}x=a, \end{cases} \] where \(N \geq 3\), \(a>0\), \(\lambda \in \mathbb{R}\), \(h \neq const\) and \(2^*=\frac{2N}{N-2}\) is the Sobolev critical exponent. In the \(L^2\)-subcritical regime (i.e. \(2<q<2+\frac{4}{N}\)), by proposing some new conditions on \(h\), we verify that the corresponding Pohozaev manifold is a natural constraint and establish the existence of normalized ground states. Compared to the \(L^2\)-subcritical regime, it is necessary to apply some reverse conditions to \(h\) provided that at least \(L^2\)-critical regime (i.e. \(2+\frac{4}{N} \leq q<2^*\)) is considered. We prove the existence of minimizer on the Pohozaev manifold of the associated energy functional and determine that the minimizer is a normalized solution by using the classical deformation lemma. In particular, by imposing further assumptions on \(h\), the ground states can be obtained.