Let $X$ be a topological space. Consider the category of (finite) covers of $X$, which we denote by $\mathrm{Cov}(X)$.
What are the group objects in $\mathrm{Cov}(X)$?
Assume that $X$ admits an universal cover and let $x_0 \in X$. Then the functor
$$\mathrm{Cov}(X)\longrightarrow \pi_1(X,x_0)\text{-Sets}, $$ associating the monodromy action on the fibre $p^{-1}(x_0)$ to a cover $p$, is an equivalence.
Under this equivalence these correspond to (finite) groups $G$ with a homomorphism $\pi_1(X,x_0)\to \mathrm{Aut}(G)$. In particular such a group object covers $p$ should have a group structure on $p^{-1}(x_0)$. Is there a connection to principal bundles?