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We investigate the existence, uniqueness, and radial symmetry of normalizedsolutions to the Schr\"{o}dinger Poisson equation with non-autonomousnonlinearity $f(x,u)$: \begin{equation} -\triangleu+(|x|^{-1}*|u|^2)u=f(x,u)+\lambda u, \nonumber \end{equation} subject to theconstraint $\mathcal{S}_c=\{u\in H^1(\mathbb{R}^3)|\int_{\mathbb{R}^3}u^2=c>0\}$. We consider three cases based on the behavior of $f(x,u)$: the $L^2$supercritical case, the $L^2$ subcritical case with growth speed less thanthree power times, and the $L^2$ subcritical case with growth speed more thanthree power times. We establish the existence of solutions using threedifferent methods depending on $f(x,u)$. Furthermore, we demonstrate theuniqueness and radial symmetry of normalized solutions using an implicitfunction framework when $c$ is small.

On existence, uniqueness and radiality of normalized solutions to Schr\"{o}dinger-Poisson equations with non-autonomous nonlinearity