This article addresses the question of which non-empty, compact, proper subsets $E$ of the extended complex plane $\widehat{\mathbb C}$ have the feature that, for some $K$ in $[1,\infty)$, the family of $K$-quasiconformal self-mappings of $\widehat{\mathbb C}$ which leave $E$ invariant acts transitively on the set $E\times E^c$, where $E^c$ is the complement of $E$ in $\widehat{\mathbb C}$. The main result in the paper asserts that the class of sets with this property comprises all one- and two-point subsets of $\widehat{\mathbb C}$, all quasicircles in $\widehat{\mathbb C}$ and all images of the Cantor ternary set under quasiconformal self-mappings of $\widehat{\mathbb C}$. It is shown that the third category includes the limit set of any non-cyclic, finitely generated Schottky group.