  We consider the stability of Sasaki-extremal metrics under deformations of
the complex structure on the Reeb foliation. Given such a deformation
preserving the action of a compact subgroup of the automorphism group of a
Sasaki-extremal structure, a sufficient condition is given involving the
nondegeneracy of the relative Futaki invariant for the deformations to contain
Sasaki-extremal structures.
  Deformations of Sasaki-Einstein metrics are also considered, where it
suffices that the deformation preserve a maximal torus. As an application, new
families of Sasaki-Einstein and Sasaki-extremal metrics are given on
deformations of well known 3-Sasaki 7-manifolds.
