  We study the uniqueness and the radial symmetry of minimizers on a
Pohozaev-Nehari manifold to the Schr\"{o}dinger Poisson equation with a general
nonlinearity $f(u)$. Particularly, we allow that $f$ is $L^2$ supercritical.
The main result shows that minimizers are unique and radially symmetric modulo
suitable translations.
