A non-injective version of Wigner's theorem

Let $H$ be a complex Hilbert space and let ${\mathcal F}_{s}(H)$ the real vector formed by all self-adjoint finite rank operators on $H$. We prove following non-injective version of Wigner's theorem: every linear operator sending one projections to (without any additional assumption) is induced or conjugate-linear isometry it constant set projections.