It is well known that if $(P, \leq)$ is a partial order then $\leq$ can always be extended to a linear order. This is sometimes called Szpilrajn´s theorem although it had been previously proved by Banach, Kuratowski and Tarski.

Now suppose that $f$ is an automorphism of $(P, \leq)$ and we want to extend $\leq$ to a linear order in such a way that $f$ remains an automorphism. Of course, this is not always possible since $f$ could have a finite orbit and automorphisms of linear orders can´t have finite orbits; but I wonder if this is the only obstruction. So let $A$ be the collection of all those $f$´s for which it is possible (for instance $Id_P \in A$); here are my questions:

**1)** Is it true that If $f$ has no finite orbits then $f \in A$?

**2)** Is $A$ a subgroup of $Aut(P, \leq)$?

and the vaguer

**3)** If the answer to 1 is negative. Can we somehow characterize the elements of $A$?

Perhaps this is all well known and studied but I couldn´t find anything at all in the literature, so references are also appreciated.