Homogeneous hyper-Hermitian metrics which are conformally hyper-Kähler

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-Kähler metric. We describe explicitely the corresponding hyper-Kähler metrics and it follows that, in four dimensions, these are the only hyper-Kähler metrics containing a homogeneous metric in its conformal class.

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