  The ground state properties of an Ising chain with nearest ($J_{1}$) and
next-nearest neighbor ($J_{2}$) interactions in a transverse field are
investigated using the density matrix renormalization group and cluster
mean-field theory methods. Its quantum phase diagram has two ordered regions
(separated by a quantum disordered phase), one of which has simple
ferromagnetic or N\'eel order depending upon the sign of $J_{1}$, and the other
is the double-staggered antiferromagnetic phase. The presence of the
Majorana-like edge-modes in these ordered phases is inferred by calculating the
end-to-end spin-spin correlations on open chains. It is found that there occur
four edge modes in the double-staggered phase, while the ferromagnetic and
N\'eel phases support two edge modes, except very near $J_{1}=0$ where it seems
they have four edge modes like the exact case at $J_{1}=0$ itself.
