Laplace-Beltrami on $SO(3)$

Question

Denoting the double cover map $SU(2)\rightarrow SO(3)$ by $\phi$, we have an induced monomorphism $C^\infty(SO(3))\rightarrow C^\infty(SU(2))$, $f\mapsto f\circ\phi$. We know that the Lie-algebra of both these groups is $\mathfrak{su}(2)$, and therefore, the Casimir operator is identical. Suppose that $f\in C^\infty(SO(3))$ satisfies $\Delta_{SO(3)}f=-\lambda f$; is it true that $\Delta_{SU(2)}(f\circ\phi)=-\lambda (f\circ\phi)$? And if so, does the Fourier transform criteria imply this (not all $SU(2)$ irrep are $SO(3)$ irrep)? I will be grateful if someone could please point to a reference where the metric property of double cover map (on spin group, in general) is treated.