Let $A,B$ be sufficiently connected spaces so as to have the category of coverings equivalent to the category of representations of the fundamental groupoid.
Given a covering map of $A$ and a continuous map $A\overset{f}{\to} B$, we may pass to the associated representation of $\pi_1A$ and take its left Kan extension along $\pi_1A\overset{\pi_1f}{\to} \pi_1B$ to obtain a representation of $\pi_1B$. Finally, we may integrate it into a covering map of $B$. This defines the pushforward of a covering map along a continuous map.
Question. How to calculate - or at least anticipate - this pushforward?
For instance, what is the pushforward of the $n$-fold cover of $\mathbb S^1$ along the $k$-fold cover of $\mathbb S^1$?
I'm trying to use the formula for pointwise Kan extensions, but I'm struggling. At least for $f:\bf 1\to \mathbb S^1$, the fibers of the pushforward of the identity on the point seem to be $\mathbb Z$, but even in this case I'm struggling to calculate the parallel transport.