I asked a question where @YCor provided a construction that seems to enable a group construction satisfying some properties when $n\ne 5$. However, in the case $n=5$, I am starting to think no such group exists. Precisely, I am asking if the following claim is true.

There is no closed subgroup $G$ of the the orthogonal group $O(5)$ along with a continuous surjective homomorphism $\phi : G\to \mathbb{Z}_2 = \{1, -1\}$ such that

1. If $x\in\mathbb{R}^n,\, x\ne 0$ then the orbit $Gx$ contains infinitely many distinct elements.
2. There exists $x\in \mathbb{R^n}$ such that $\{g\in G : gx = x\}\subseteq \operatorname{ker}\phi$.

Any ideas on how to approach this are appreciated.