Consider a torus with three handles, where one handle is much larger than the others. Obviously there is a smooth and free $\mathbb{Z}_3$ action on the manifold.

Let $\gamma$ be a small geodesic loop going through one of the small handles. Let $z\gamma$ be an image of $\gamma$ under the group action which goes through the large handle. Let $p$ be a point on $\gamma$ such that $zp$ is far out on the large handle.

Now the distance between $p$ and $zp$ is almost the diameter of the manifold, and there is no third point $z^2p$ which makes an equilateral triangle with them. So there is no smooth and equilateral $\mathbb{Z}_3$ action on the manifold.

Consider a torus with three handles, where one handle is much larger than the others, and with a smooth and free $\mathbb{Z}_3$ action which permutes the handles.

Let $\gamma$ be a small geodesic loop going through one of the small handles. Let $z\gamma$ be an image of $\gamma$ under the group action which goes through the large handle. Let $p$ be a point on $\gamma$ such that $zp$ is far out on the large handle.

Now the distance between $p$ and $zp$ is almost the diameter of the manifold, and there is no third point $z^2p$ which makes an equilateral triangle with them. So there is no smooth and equilateral $\mathbb{Z}_3$ action on the manifold.