Monopole Floer homology is a powerful invariant of in 3-dimensional topology. By its naturality, monopole Floer homology of a 3-manifold is acted upon by its mapping class group. In a recent work with Danny Ruberman and Nikolai Saveliev, we make the first step towards understanding this action by calculating the Lefschetz numbers of certain involutions making the 3–manifold into a double branched cover over a link in the 3–sphere. Various applications of of this formula will be discussed.