Is there a standard name for a category $C$ in which coproducts exist, and in which coequalizers exist for any action of a finite group on an object of $C$? Equivalently, colimits are required to exist only for diagrams of the form $I \to C$ where $I$ is a finite groupoid.

Some examples satisfying this condition which are not cocomplete: the category of orbifolds, and Grothendieck's category of motives. The latter is closed under quotients of finite groups since it is $\mathbf Q$-linear and pseudo-abelian.