We use the two-dimensional Schwinger model to investigate how lattice fermions perceive the global topological charge $q\in\mathbf{Z}$ of a given gauge background $U$. After a warm-up part devoted to staggered, Adams, Wilson and naive fermions, we consider Karsten-Wilczek and Borici-Creutz fermions, which are in the class of minimally doubled lattice fermion actions. We focus on the eigenvalue spectrum and the chiralities of the pertinent eigenmodes. Without modification both minimally doubled actions are found to be insensitive to topology, but in either case it is possible to define a suitable species-splitting term to make the resulting operator topology aware.