  In this paper we study a version of the Hermitian curvature flow (HCF). We
focus on complex homogeneous manifolds equipped with induced metrics. We prove
that this finite-dimensional space of metrics is invariant under the HCF and
write down the corresponding ODE on the space of Hermitian forms on the
underlying Lie algebra. Using these computations we construct HCF-Einstein
metrics on $G$-homogeneous manifolds, where $G$ is a complexification of a
compact simple Lie group. We conjecture that under the HCF any induced metric
on such a manifold pinches towards the HCF-Einstein metric. For a nilpotent or
solvable complex Lie group equipped with an induced metric we investigate the
blow-up behavior of the HCF.
