Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. Let $f:X\to M_{n}(\mathbb{C})$ be a continuous function. Under what of the following two conditions we can say:

There exist $x_{0}\in X$ such that the following integral is non invertible:

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

Condition 1: $G$ is abelian and $\rho$ is a non trivial representation.

Condition 2: $\rho$ is an irreducible representation.

This question is motivated by this post and proposition $3$ of this note.