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 $xyx^{-1}y^{-1}$ is closed under the $G$-action but is not a free $G$-group.

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.

Let $G$ have two elements; the free group on two generators $x,y$ with the action of $G$ interchanging them is a free $G$-group with one generator. Its subgroup generated by $xyx^{-1}y^{-1}$ is closed under the $G$-action but is not a free $G$-group.

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 $xyx^{-1}y^{-1}$ is closed under the $G$-action but is not a free $G$-group.