Let $w$ be a word in free group $F$ on finitely many generators. We will look at $w$ as word map on groups. It is clear that there exists an endomorphism $\phi$ of $F$ such that $\phi(w) = w^{-1}$ if and only if for every group $G$, the image of word map ($Im(w)$) is closed with respect to inverse. My question is if $w$ has a property which is for every finite group $G$, if $g \in Im(w)$ then $g^{-1} \in Im(w)$, then does there exist an endomorphism $\theta$ of free group $F$ such that $\theta (w) = w^{-1}$.

Let $w$ be a word in free group $F$ on finitely many generators. We will look at $w$ as word map on groups. It is clear that there exists an endomorphism $\phi$ of $F$ such that $\phi(w) = w^{-1}$ if and only if for every group $G$, the image of word map ($\mathrm{Im}(w)$) is closed with respect to inverse. My question is if $w$ has a property which is for every finite group $G$, if $g \in \mathrm{Im}(w)$ then $g^{-1} \in \mathrm{Im}(w)$, then does there exist an endomorphism $\theta$ of free group $F$ such that $\theta (w) = w^{-1}$.