Equivariant vector bundle structure on the tangent bundle of compact Riemann surfaces with non trivial action on the base space,

Question

Let $M_{g}$ be the compact Riemann surface with $g\geq 2$. Is there an infinit group $G$ with an equivariant action on the pair $(TM_{g}, M_{g})$ such that the action on the fibers preserves the inner product, moreover there is no a $G$-fixed point on the base space. Recall that a $G$-fixed point is a point $p$ with $g.p=p$ for all $g\in G$.

This question is a remedy of my previous post bellow

https://mathoverflow.net/questions/252271/equivariant-bundle-structure-on-the-tangent-bundle-of-compact-riemann-surfaces