  We establish weighted $L^p$-Fourier-extension estimates for $O(N-k) \times
O(k)$-invariant functions defined on the unit sphere $\mathbb{S}^{N-1}$,
allowing for exponents $p$ below the Stein-Tomas critical exponent
$\frac{2(N+1)}{N-1}$. Moreover, in the more general setting of an arbitrary
closed subgroup $G \subset O(N)$ and $G$-invariant functions, we study the
implications of weighted Fourier-extension estimates with regard to boundedness
and nonvanishing properties of the corresponding weighted Helmholtz resolvent
operator. Finally, we use these properties to derive new existence results for
$G$-invariant solutions to the nonlinear Helmholtz equation
  $$ - \Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}), $$
where $Q$ is a nonnegative bounded and $G$-invariant weight function.
