This is a question of terminology. I want to talk about the category whose...
- ...objects are pairs $(G,M)$, where $G$ is a group and $M$ is a $G$-set.
- ...morphisms $(G,M)\rightarrow (G',M')$ are pairs $(f_G,f_M)$, where $f_G:G\rightarrow G'$ is a group homomorphism, and $f_M:M\rightarrow M'$ is a set map such that $$ f_M(g\cdot m) = f_G(g)\cdot f_M(m)$$ for all $g\in G$ and $m\in M$.
I've been calling these decorated groups (and their morphisms), since they've been arising in connection with decorated local systems. However, I'd prefer a more standard name, hopefully one which evokes the correct idea before explanation.