Stability of properties of locales under groups

Given a particular collection of categorical axioms, aimed at capturing properties of the category of locales, we show that if $\mathcal{C}$ is a category that satisfies the axioms then so too is the category $[ G, $\mathcal{C}$]$of$G$-objects,foranyinternalgroup$G$. To achieve this we prove a general categorical result: if an object $S$ is double exponentiable in a category with finite products then so is its associated trivial $G$-object $(S,{\pi }_2:G\times S\to S)$. The result holds even if $S$ is not exponentiable. An example is given of a category $\mathcal{C}$ that satisfies the axioms, but for which there is no elementary topos $\mathcal{E}$ such that $\mathcal{C}$ is the category of locales over $\mathcal{E}$. It is shown, in outline, how the results can be extended from groups to groupoids.