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.

`$G$`

and then refer to "the category of`$G$`

-sets" (with obvious morphisms). What is wanted here seems to come up rarely in practice: letting the group as well as the set vary arbitrarily. I'd be content just to refer to the category as before, but with no specific`$G$`

fixed. I also have no objection to "the category of group actions" (whereas "the category of actions" is really too vague). Until mathematicians elect a dictator to decide such things, we'll probably all do our own thing anyway. – Jim Humphreys May 12 '11 at 17:21