  Let (X,\Omega) be a closed polarized complex manifold, g be an extremal
metric on X that represents the K\"ahler class \Omega, and G be a compact
connected subgroup of the isometry group Isom(X,g). Assume that the Futaki
invariant relative to G is nondegenerate at g. Consider a smooth family $(M \to
B)$ of polarized complex deformations of (X,\Omega)\simeq (M_0,\Theta_0)
provided with a holomorphic action of G with trivial action on B. Then for
every t\in B sufficiently small, there exists an h^{1,1}(X)-dimensional family
of extremal Kaehler metrics on M_t whose K\"ahler classes are arbitrarily close
to \Theta_t. We apply this deformation theory to show that certain complex
deformations of the Mukai-Umemura 3-fold admit Kaehler-Einstein metrics.
