Let $g$ a finite-dimensional complex simple Lie algebra and $\sigma$ a finite order Dynikin diagram automorphism of $g$.

Consider $\tilde g$ as the loop algebra associated to $g$, and $\tilde g^\sigma$ as the twisted affine Lie algebra (associated to $g$) in the spirit of the theory developed in the book ''Infinite dimensional Lie algebra" by Kac, or ''Lie algebras of Finite and Affine Type'' by Roger Carter.

It is easy to see that given an action of $\tilde g$ we have an action of $\tilde g^\sigma$ which is given by the restriction of the action of $\tilde g$. On the opposite way, my question is: Given an action of $\tilde g^\sigma$ on a module $M$, is it possible to extend this action to $\tilde g$?

Moreover, if we restrict to the context of universal highest-weight modules for $\tilde g^\sigma$ can we produce a highest-weight module for $\tilde g$ by extending the action of $\tilde g^\sigma$?

What should be a reference for this subject concerning about these extensions?

Thanks,