Edit: According to the comment of Andreas Cap I revise the integral formula in the question.

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 which of the following two conditions we can say:

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

$$ \int_{G} \rho(g)f(g^{-1}.x_{0})d\mu $$

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

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

In particular is the following statement true:

Assume that $\phi$ is a order $n$ homeomorphism on $S^{4}$ and $\lambda \neq 1$ is a quaternion with $\lambda^{n}=1$.Assume that $f:S^{4}\to \mathbb{R}^{4} \simeq \mathbb{H}$ is a continuous function. then there is a point $x_{0}\in S^{4}$ such that $\sum_{i=0}^{n-1} \lambda^{n-i}f(\phi^{i}(x_{0}))=0$.

Note that the particular case $\lambda=-1$ and $\phi=$ The antipodal map is the classical Borsuk-Ulam theorem.

The motivation for the last part of the question is that the above statement is true if we replace $S^{4}$ and $\mathbb{R}^{4}$ by $S^{2}$ and $\mathbb{R}^{2}\simeq \mathbb{C}$, respectively and choose $\lambda \in \mathbb{C}$.

Note: To obtain a particular result we can put $X=S^{2}$ and assume that $G$ is a compact Lie group. So the answer to the first question is affirmative provided that we have a positive answer to the following question:

Question: Let $A$ be the $C^{*}$ algebra of all continuous functions from $S^{2}$ to $M_{n}(\mathbb{C})$. We now that all invertible elements are homotop to identity. Is it true to say that every invertible element $a$ has a logarithm $b$ where $b$ can be represented as a power series in $a$?