  Consider a fibred compact K\"ahler manifold X endowed with a relatively ample
line bundle, such that each fibre admits a constant scalar curvature K\"ahler
metric and has discrete automorphism group. Assuming the base of the fibration
admits a twisted extremal metric where the twisting form is a certain
Weil-Petersson type metric, we prove that X admits an extremal metric for
polarisations making the fibres small. Thus X admits a constant scalar
curvature K\"ahler metric if and only if the Futaki invariant vanishes. This
extends a result of Fine, who proved this result when the base admits no
continuous automorphisms.
  As consequences of our techniques, we obtain analogues for maps of various
fundamental results for varieties: if a map admits a twisted constant scalar
curvature K\"ahler metric metric, then its automorphism group is reductive; a
twisted extremal metric is invariant under a maximal compact subgroup of the
automorphism group of the map; there is a geometric interpretation for
uniqueness of twisted extremal metrics on maps.
