  We consider combined power-type nonlinear scalar field equations with the
Sobolev critical exponent. In \cite{AIKN3}, it was shown that if the frequency
parameter is sufficiently small, then the positive ground state is
nondegenerate and linearly unstable, together with an application to a study of
global dynamics for nonlinear Schr\"odinger equations. In this paper, we prove
the nondegeneracy and linear instability of the ground state frequency for
sufficiently large frequency parameters. Moreover, we show that the derivative
of the mass of ground state with respect to the frequency is negative.
