Is there a natural number $n$, a compact Lie group $G$ of dimension less than $n$ and a continuous map $f:S^n \to G$ with $f(-x)=f(x)^{-1}$ such that $f$ is not a null homotopic map? This question was included in the following MSE post but I did not received any answer.

https://math.stackexchange.com/questions/2757987/existence-of-a-certain-equivariant-map-from-the-sphere-to-a-compact-lie-group-of

Is there a natural number $n$, a compact Lie group $G$ of dimension less than $n$ and a continuous map $f:S^n \to G$ with $f(-x)=f(x)^{-1}$, such that $f$ is not a null homotopic map? This question was included in this MSE post but I did not receive any answer.