Representability of permutation representations on coalgebras and the isomorphism problem

Let $G$ be a group and let $\rho \text{\hspace{0.17em}⁣}:G\to \mathrm{Sym}(V)$ be a permutation representation of $G$ on a set $V$. We prove that there is a faithful $G$-coalgebra $C$ such that $G$ arises as the image of the restriction of $\mathrm{Aut}(C)$ to $G(C)$, the set of grouplike elements of $C$. Furthermore, we show that $V$ can be regarded as a subset of $G(C)$ invariant through the $G$-action, and that the composition of the inclusion $G\hookrightarrow \mathrm{Aut}(C)$ with the restriction $\mathrm{Aut}(C)\to \mathrm{Sym}(V)$ is precisely $\rho$. We use these results to prove that isomorphism classes of certain families of groups can be distinguished through the coalgebras on which they act faithfully.