  The study of extremal properties of the spectrum often involves restricting
the metrics under consideration. Motivated by the work of Abreu and Freitas in
the case of the sphere $S^2$ endowed with $S^1$-invariant metrics, we consider
the subsequence $\lambda_k^G$ of the spectrum of a Riemannian manifold $M$
which corresponds to metrics and functions invariant under the action of a
compact Lie group $G$. If $G$ has dimension at least 1, we show that the
functional $\lambda_k^G$ admits no extremal metric under volume-preserving
$G$-invariant deformations. If, moreover, $M$ has dimension at least three,
then the functional $\lambda_k^G$ is unbounded when restricted to any conformal
class of $G$-invariant metrics of fixed volume. As a special case of this, we
can consider the standard O(n)-action on $S^n$; however, if we also require the
metric to be induced by an embedding of $S^n$ in $\mathbb{R}^{n+1}$, we get an
optimal upper bound on $\lambda_k^G$.
