Let $f \colon (Y,y_0) \to (X,x_0)$ be a finite-to-one pointed covering map. The pushforward gives an inclusion $f_* \colon \pi_1(Y,y_0) \to \pi_1(X,x_0)$. If we take the universal cover $\widetilde{X}$ of $X$ and mod out by the action of the image $f_*(\pi_1(Y,y_0))$, we get a cover of $X$ that is naturally isomorphic to $Y$. If $f$ is an abelian cover (i.e., normal, such that the group of deck transformations is abelian), then we can replace $\pi_1(-)$ by $H_1(-,\mathbb{Z})$ and this all goes through.
This is in analogy to the situation in Galois theory: Let $L/K$ be a finite field extension. The absolute Galois group $G_L$ is a subgroup of $G_K$, since $\overline{L} = \overline{K}$, and any automorphism that fixes $L$ will also fix $K$. If the extension is normal, we have $G_K/G_L \cong \operatorname{Gal}(L/K)$. (In this analogy, the inclusion $G_L \to G_K$ is the pushforward $f_* \colon \pi_1(Y,y_0) \to \pi_1(X,x_0)$.)
The questions:
What happens when we relax the condition that $f$ is finite-to-one? I know that, in the field extension case, we get the extra information that $G_L$ is a closed subgroup of $G_K$ in the profinite topology; this is what allows $G_K/G_L$ to inherit the topology of $G_K$. Should I think that there is some notion of a profinite covering map, and that the theory extends just like it does in the case of fields with algebraic extensions? Specifically, can I say that $f_* \colon \pi_1(Y,y_0) \to \pi_1(X,x_0)$ is injective, its image is a closed, normal subgroup, and that the quotient $\pi_1(X,x_0)/f_*(\pi_1(Y,y_0))$ is isomorphic to the group of deck transformations $\operatorname{Aut}(Y/X)$ as topological groups?
Is there a good reference I can give here? The infinite case is important for my application, specifically the profinite case. Should I just cite somewhere in Grothendieck?
