  On a smooth closed oriented 4-manifold $M$ with a smooth action by a finite
group $G$, we show that a $G$-monopole class gives the $L^2$-estimate of the
Ricci curvature of a $G$-invariant Riemannian metric, and derive a topological
obstruction to the existence of a $G$-invariant nonsingular solution to the
normalized Ricci flow on $M$.
  In particular, for certain $m$ and $n$, $m\Bbb CP_2 # n\bar{\Bbb CP}_2$
admits an infinite family of topologically equivalent but smoothly distinct
non-free actions of $\Bbb Z_d$ such that it admits no nonsingular solution to
the normalized Ricci flow for any initial metric invariant under such an
action, where $d>1$ is a non-prime integer.
  We also compute the $G$-Yamabe invariants of some 4-manifolds with
$G$-monopole classes and the oribifold Yamabe invariants of some 4-orbifolds.
