To any action of a finite group $ G $ on a closed symplectic $ 4 $-manifold $ ( M , \omega ) $, one can associate a symplectic resolution $ \pi : ( \widetilde{M} , \widetilde{\omega} ) \to M / G $. When $ b_+^G ( M ) \geq 2 $, equivariant Seiberg-Witten-Taubes theory implies the existence of an invariant collection $ \mathcal{K} $ of pseudoholomorphic curves representing $ c_1 ( K_\omega ) $ and containing (almost) all the fixed points, and when $ ( M , \omega ) $ is minimal with $ c_1 ( K_\omega )^2 = 0 $ the possibilities for $ \mathcal{K} $ and its symmetries are constrained. In this work, we explore the nature of $ ( \widetilde{M} , \widetilde{\omega} ) $ and its minimal model when $ ( M , \omega ) $ has symplecitc Kodaira dimension $ \kappa^s ( M , \omega ) = 1 $ and $ G = \Z / p $ for $ p $ prime by tracing the evolution of such a collection $ \mathcal{K} $ through the quotient $ M \to M / G $ and resolution $ \widetilde{M} \to M / G $, finally to the minimal model. We apply this to address a conjecture of Chen, showing for $ G = \Z / p $ that if $ \kappa^s ( M , \omega ) = 1 $ then $ \kappa^s ( \widetilde{M} , \widetilde{\omega} ) \in \{ 0 , 1 \} $.