**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 what of the  following two  conditions we can say:

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

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


**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 statment is true if we replace $S^{4}$ and $\mathbb{R}^{4}$  by $S^{2}$ and $\mathbb{R}^{2}\simeq \mathbb{C}$, respectively  and choose $\lambda \in \mathbb{C}$.