  We investigate the fraction of metal nuclei in the relativistic jets of
gamma-ray bursts associated with core-collapse supernovae. We simulate the
fallback in jet-induced explosions with two-dimensional relativistic
hydrodynamics calculations and the jet acceleration with steady, radial,
relativistic magnetohydrodynamics calculations, and derive detail nuclear
composition of the jet by postprocessing calculation. We found that if the
temperature at the jet launch site is above $4.7\times 10^9$K,
quasi-statistical equilibrium (QSE) is established and heavy nuclei are
dissociated to light particles such as $^4$He during the acceleration of the
jets. The criterion for the survival of metal nuclei is written in terms of the
isotropic jet luminosity as $L_{\rm j}^{\rm iso} \lesssim 3.9\times
10^{50}(R_{\rm i}/10^7{\rm cm})^2 (1+\sigma_{\rm i})~{\rm erg~s^{-1}}$, where
$R_{\rm i}$ and $\sigma_{\rm i}$ are the initial radius of the jets and the
initial magnetization parameter, respectively. If the jet is initially
dominated by radiation field (i.e., $\sigma_{\rm i} \ll 1$) and the isotropic
luminosity is relatively high ($L_{\rm j}^{\rm iso}\gtrsim 4\times 10^{52}~{\rm
erg s^{-1}}$), the metal nuclei cannot survive in the jet. On the other hand,
if the jet is mainly accelerated by magnetic field (i.e., $\sigma_{\rm i}
\gg1$), metal nuclei initially contained in the jet can survive without serious
dissociation even for the case of high luminosity jet. If the jet contains
metal nuclei, the dominant nuclei are $^{28}$Si, $^{16}$O, and $^{32}$S and the
mean mass number can be $\langle$A$\rangle\sim25$.
