We consider a discrete sine-Gordon equation with a phase shift. We analyze the stability of a lattice kappa kink which is the ground state of the system. The dependence of the localized mode frequency of a kappa kink on the topological charge is analyzed numerically and analytically. We show that there is a certain range of parameter values for the coupling constant and the phase shift for an internal mode to exist. A semianalytical calculation on the existence of an internal mode of a fractional kink is presented using the characteristic that a long wavelength phonon is perfectly transmitted at the appearance of an internal mode. We also briefly discuss another type of fractional kink that exists for any value of coupling constant. This fractional kink is stable in the weak coupling limit but unstable in the opposite one.