Summary: We answer four questions from a recent paper of S. Rao and I. Shinkar [Comb. Probab. Comput. 27, No. 3, 411–426 (2018; Zbl 1392.68319)] on Lipschitz bijections between functions from \(\{0, 1\}^n\) to \(\{0, 1\}\). (1) We show that there is no \(O(1)\)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on \(O(1)\) input bits. (2) We give a construction for a mapping from XOR to Majority which has average stretch \(O(\sqrt{n})\), matching a previously known lower bound. (3) We give a 3-Lipschitz embedding \(\phi \colon \{0,1\}^n \to \{0,1\}^{2n+1}\) such that \({\mathrm{XOR}}(x) = {\mathrm{Majority}}(\phi (x))\) for all \(x \in \{0,1\}^n\). (4) We show that with high probability there is an \(O(1)\)-bi-Lipschitz mapping from Dictator to a uniformly random balanced function.