On a smooth closed oriented 4-manifold $$M$$ M with a smooth action of a finite group $$G$$ G on a Spin $$^c$$ c structure, $$G$$ G -monopole invariant is defined by “counting” $$G$$ G -invariant solutions of Seiberg–Witten equations for any $$G$$ G -invariant Riemannian metric on $$M$$ M . We compute $$G$$ G -monopole invariants on some $$G$$ G -manifolds. For example, the connected sum of $$k$$ k copies of a 4-manifold with nontrivial mod 2 Seiberg–Witten invariant has nonzero $$\mathbb Z_k$$ Z k -monopole invariant mod 2, where the $$\mathbb Z_k$$ Z k -action is given by cyclic permutations of $$k$$ k summands.