Groups in which every Lagrange subset is a factor

We determine the finite groups $G$ in which every subset $A\subseteq G$ of cardinality dividing the order of $G$ is a \emph{factor}, i.e. has a complement $B\subseteq G$ of cardinality $\mathrm{\mid }G\mathrm{\mid }\mathrm{/}\mathrm{\mid }A\mathrm{\mid }$ such that $G=A\cdot B$ or $G=B\cdot A$.