  It is known that if each point $x$ of a dynamical system is generic for some
invariant measure $\mu_x$, then there is a strong connection between certain
ergodic and topological properties of that system. In particular, if the acting
group is abelian and the map $x\mapsto \mu_x$ is continuous, then every orbit
closure is uniquely ergodic.
  In this note, we show that if the acting group is not abelian, orbit closures
may well support more than one ergodic measure even if $x\mapsto \mu_x$ is
continuous. We provide examples of such a situation via actions of the group of
all orientation preserving homeomorphisms on the unit interval as well as the
Lamplighter group. Moreover, we generalize several results concerning weakly
mean equicontinuous group actions beyond the setting of countable discrete
abelian/amenable groups.
