The Nielsen-Schreier theorem states that subgroups of a free subgroup are free.
Is this hold also for groups with operations?


Explicitly, let $G$ be a fixed group. Let $F$ be a group with $G$-action which is free (as a group with $G$-action). Let $F'\subset F$ be a subgroup closed by the $G$-action. Then, must $F$ be free as a group with $G$-action?



Nielsen-Schreier with operations
Let $G$ be a two element group; the free group on two generators $x,y$ with the action of $G$ interchanging them is a free $G$-group (on one generator). Its subgroup generated by $xy^{-1}$ is closed under the $G$-action but is not a free $G$-group.


