  The systematic study of harmonic self-maps on cohomogeneity one manifolds has
recently been initiated by P\"uttmann and the second named author in
\cite{MR4000241}. In this article we investigate the corresponding Jacobi
equation describing the equivariant stability of such harmonic self-maps.
Besides several general statements concerning their equivariant stability we
explicitly solve the Jacobi equation for some harmonic self-maps in the cases
of spheres, special orthogonal groups and $\SU(3)$. In particular, we show by
an explicit calculation that for specific cohomogeneity one actions on the
sphere the identity map is equivariantly stable.
