Fix a nice complex algebraic variety $X$ with base point $x$. (We will work in the analytic topology.)

Let $G$ be a finite group. Then the set of normal subgroups of $\pi_1(X,x)$ with quotient $G$ corresponds to the set of $X$-torsors for $G$. (We write $G$ for the constant sheaf on $X$ associated to $G$.)

This is just saying that **Galois** covers of $X$ correspond to torsors for $G$.

Now, not all covers of $X$ are Galois. In general, finite covers of $X$ correspond (in some sense) to finite index subgroups of $\pi_1(X,x)$ and the degree is given by the index.

I'd like to know if we can replace the set $H^1(X,G)$ of torsors for $G$ by something "torsorish" and still have a correspondence with the set of finite index subgroups of $\pi(X,x)$ of some fixed degree.

connectedcovers of degree $n$...' If you don't require connectedness, you can also drop the transitivity assumption. – HJRW Mar 18 '12 at 19:31