  Let $g$ be a hyper-Hermitian metric on a simply connected hypercomplex
four-manifold $M$. We show that when the isometry group $I(M,g)$ contains a
subgroup acting simply transitively on $M$ by hypercomplex isometries then the
metric $g$ is conformal to a hyper-K\"ahler metric. We describe explicitely the
corresponding hyper-K\"ahler metrics and it follows that, in four dimensions,
these are the only hyper-K\"ahler metrics containing a homogeneous metric in
its conformal class.
