Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected topological space $X$ and $\rho$ is a unitary irreducible representation of $G$. Let $f:X\to \mathbb{C}^{n}$ be a continuous function.

Is it true to say that there exist $x_{0}\in X$ such that the following integral is non invertible:

$$ \int_{G} \rho(g)f(g.x_{0}))d\mu $$

This question is motivated by Integrals of representations over geodesics

and proposition $3$ of http://arxiv.org/abs/1110.0091