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?

characterizationis: a $G$-group is free (as $G$-group) if and only if it admits a subset $X$ such that $X$ is a free $G$-subset, and $G$ is freely generated by $X$ as group. $\endgroup$10more comments