  For any $G$-invariant metric on a compact homogeneous space $M=G/K$, we give
a formula for the Lichnerowicz Laplacian restricted to the space of all
$G$-invariant symmetric $2$-tensors in terms of the structural constants of
$G/K$. As an application, we compute the $G$-invariant spectrum of the
Lichnerowicz Laplacian for all the Einstein metrics on most generalized Wallach
spaces and any flag manifold with $b_2(M)=1$. This allows to deduce the
$G$-stability and critical point types of each of such Einstein metrics as a
critical point of the scalar curvature functional.
