On extendability of permutations

Let $V$ be a left vector space over a division ring and let ${\mathcal P}(V)$ be the associated projective space. We describe all finite subsets $X\subset V$ such that every permutation on $X$ can be extended to a linear automorphism of $V$ and all finite subsets ${\mathcal X}\subset {\mathcal P}(V)$ such that every permutation on ${\mathcal X}$ can be extended to an element of ${\rm PGL}(V)$. Also, we reformulate the results in terms of linear and projective representations of symmetric groups.