  We study the quantum fidelity (groundstate overlap) near quantum phase
transitions of the Ising universality class in one dimensional (1D) systems of
finite size L. Prominent examples occur in magnetic systems (e.g. spin-Peierls,
the anisotropic XY model), and in 1D topological insulators of any
topologically nontrivial Altland-Zirnbauer-Kitaev universality class. The
rescaled fidelity susceptibility is a function of the only dimensionless
parameter LM, where 2M is the gap in the fermionic spectrum. We present
analytic expressions for the fidelity susceptibility for periodic and open
boundaries conditions with zero, one or two edgestates. The latter are shown to
have a crucial impact and alter the susceptibility both quantitatively and
qualitatively. We support our analytical solutions with numerical data.
