  We investigate fixed point properties for isometric actions of topological
groups on a wide class of metric spaces, with a particular emphasis on Hilbert
spaces. Instead of requiring the action to be continuous, we assume that it is
``controlled", i.e. compatible with respect to some natural left-invariant
coarse structure. For locally compact groups, we prove that these coarse fixed
point properties are equivalent to the usual ones, defined for continuous
actions. We deduce generalisations of two results of Gromov originally stated
for discrete groups. For Polish groups with bounded geometry (in the sense of
Rosendal), we prove a version of Serre's theorem on the stability of coarse
property FH under central extensions. As an application we prove that the group
$\text{Homeo}^+_{\mathbb Z}(\mathbb R)$ has property FH. Finally, we
characterise geometric property (T) for sequences of finite Cayley graphs in
terms of coarse property FH of a certain group.
