In this paper, we study the existence of normalized solutions for nonautonomous Kirchhoff equation involving Sobolev critical exponent \(\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\mathbb {R}^{3}}|\nabla u|^2\textrm{d}x\right) \Delta u=\lambda u+h(x)|u|^{q-2}u+|u|^4u& \text{ in }\ \mathbb {R}^3, \\ \int _{\mathbb {R}^3}|u|^2dx=c,\\ \end{array} \right. \end{aligned}\) where \(2<q<6\) , \(a,b,c>0\) , \(\lambda \in \mathbb {R}\) and \(h\in C(\mathbb {R}^{3},\mathbb {R}^+)\) . For \(2<q<\frac{10}{3}\) , we establish the existence of an interior local minimizer of constraint functional provided that \(h(x)\ge h_{\infty }=\lim _{|x|\rightarrow \infty }h(x)>0\) and prove that this minimizer is a normalized ground state if \(c>0\) is small. Moreover, we found an interesting result that if appropriate conditions are applied to ensure that the Pohozaev manifold is a natural constraint, then the corresponding minimization problem can only be achieved when \(h=h_\infty >0\) . For \(\frac{14}{3}\le q<6\) , we can prove the minimization problem is achieved and the minimizer is a normalized ground state. This result allows \(h\not =const\) , which is different from the case of \(2<q<\frac{10}{3}\) . Furthermore, some asymptotic properties are established.
