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Let $X$ be a smooth, compact, oriented $4$-manifold. Building upon work ofLi-Liu, Ruberman, Nakamura and Konno, we consider a families version ofSeiberg-Witten theory and obtain obstructions to the existence of certain groupactions on $X$ by diffeomorphisms. The obstructions show that certain groupactions on $H^2(X , \mathbb{Z})$ preserving the intersection form can not belifted an action of the same group on $X$ by diffeomorphisms. Using ourobstructions, we construct numerous examples of group actions which can berealised continuously but can not be realised smoothly for any differentiablestructure. For example, we construct compact simply-connected $4$-manifolds $X$and involutions $f : H^2(X , \mathbb{Z}) \to H^2(X , \mathbb{Z})$ such that $f$can be realised by a continuous involution on $X$ or by a diffeomorphism, butnot by an involutive diffeomorphism for any smooth structure on $X$.

Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory