Consider the following category:
- objects are triples $(G,A,\alpha)$ where $G$ is a group, $\alpha : G\times A\to A$ is a left action of $G$ on a set $A$;
- morphisms $(G,A,\alpha)\to (H,B,\beta)$ are pairs $u : G\to H$ (an homomorphism of groups) and $f : A\to B$ (a function) such that $f(\alpha(g,a))=\beta(ug,fa)$.
Composition is $(u,f)(v,f')=(uv,ff')$.
Call this category $Grp\ltimes Set$; a slick way to define it is the following: consider the (pseudo)functor $a : Grp \to Cat$ sending $G$ to the category $Set^G$ of left $G$-sets. Perform the Grothendieck construction on $a$, you get a (cloven) fibration $\begin{smallmatrix}Grp\ltimes Set\\\\ \downarrow\\\\ Grp\end{smallmatrix}$ projecting on the first component, of which $Grp\ltimes Set$ is the total category.
Now, consider the functor $r : Grp \to Grp\ltimes Set$ defined sending $G$ to $(G,G,reg)$, where $reg$ is the left regular representation of $G$, i.e.$a_g(h):=gh$. (It is easy to see that this is a functor.)
Does $r$ have a left adjoint?
A bit of context for you to understand what is my motivation in this question: consider, instead, a slightly modified version of the above construction, where actions are by group homomorphisms; then the functor $c : G\mapsto (G,G,conj)$, where $conj$ is the conjugation action, does have a left adjoint: the typical object $(G,H,\psi :G\times H\to H)$ goes to the semidirect product $G\ltimes_\psi H$.