I've come upon this MO post, and I was wondering if there was a way to generalize what is said in the comment at the very bottom. Specifically, we have that given continuous maps $f: \mathbb{S}^n \to \mathbb{S}^n$, and $g: \mathbb{S}^n \to \mathbb{R}^n$, if $f$ is an involution, then there is a point $x \in \mathbb{S}^n$ such that $g(x) = g(f(x))$, i.e. the value at $x$ is fixed under the transformation of the sphere by $f$.

We know from the answer of the post that for a general $f$, there need not be such a fixed point. But now what if instead $f$ is such that there exists a $k \in \mathbb{N}$ where $f^k$ is the identity map, do we still get such a fixed point? Furthermore, do we know about other spaces besides the sphere that has this property when $f$ is an involution?