Radial extensions of bilipschitz maps between unit spheres

Let $E_1$ and $E_2$ be real inner product spaces, and let $S_1$ and $S_2$ be the corresponding unit spheres. We consider different proofs showing that the radial extension of an $L$-bilipschitz map $f\colon S_1\to S_2$ is $L$-bilipschitz with the same constant $L$. We also consider certain other sets having this kind of an extension property.