**Motivation**

For a fixed group $G$, consider the category of $G$-groups, whose objects consist of (not necessarily abelian) groups $A$ together with an action $G\to \mathrm{Aut}(A)$. Now we can construct a nonabelian $H^1$, a functor from $G$-groups to pointed sets by mimicking the abelian construction. This construction is very common and there are Čech cohomology variants for instance. (Other variants of nonabelian cohomology $H^n(G,*)$ for all $n$ have also been constructed via crossed modules; see Labesse et al. in the Asterisque *"Cohomologie, stabilisation et changement de base"*.)

One approach to try and construct higher $H^n(G,-)$ functors for the category of $G$-groups, would be to find a similar "nonabelian bar resolution". In the abelian case, the forgetful functor $U:G\text{-Mod}\to\text{Ab}$ has a left-adjoint $F:\text{Ab}\to G\text{-Mod}$ given by $A\mapsto \mathbb{Z}G\otimes_{\mathbb{Z}} A$. In this case we can then use the corresponding cotriple to construct the bar resolution in the usual fashion.

**The Question:**

A naive thing to do would be to look for a left adjoint to the forgetful functor $U:G\text{-Group}\to\text{Group}$, sending a $G$-group $A$ to the group $A$, forgetting the $G$-action. **So my question is, does this forgetful functor have a left-adjoint? If so, what is it?** My guess would be no, but I'm not sure. I've tried various things with free products but I can neither construct one nor prove it doesn't exist.