  On a smooth closed oriented $4$-manifold $M$ with a smooth action by a
compact Lie group $G$, we define a $G$-monopole class as an element of
$H^2(M;\Bbb Z)$ which is the first Chern class of a $G$-equivariant Spin$^c$
structure which has a solution of the Seiberg-Witten equations for any
$G$-invariant Riemannian metric on $M$.
  We find $\Bbb Z_k$-monopole classes on some $\Bbb Z_k$-manifolds such as the
connected sum of $k$ copies of a 4-manifold with nontrivial mod 2
Seiberg-Witten invariant or Bauer-Furuta invariant, where the $\Bbb Z_k$-action
is a cyclic permutation of $k$ summands.
  As an application, we produce infinitely many exotic non-free actions of
$\Bbb Z_k\oplus H$ on some connected sums of finite number of $S^2\times S^2$,
$\Bbb CP_2$, $\overline{\Bbb CP}_2$, and $K3$ surfaces, where $k\geq 2$, and
$H$ is any nontrivial finite group acting freely on $S^3$.

Geometric Topology;Differential Geometry