Whenever we have a continuous map of topological spaces $f: X \to Y$ and a sheaf $\mathcal{F}$ on $Y$ (of abelian groups for example), we get an induced pullback map
$$ H^n(Y, \mathcal{F}) \to H^n(X, f^* \mathcal{F}), $$
which is described for example in Iversen, *Cohomology of Sheaves*, II.5.1. I would like to have an explicit description of this map in one particular case.

Suppose $X = Y = S^1$ and further suppose that the sheaf is a local system $\mathcal{L}$. Then its pullback $f^* \mathcal{L}$ is also a local system. We have a nice description of local systems in terms of representations of the fundamental group, and the pullback is easily describable in this language: if $$ \pi_1(Y) \to \mathrm{Aut}(\mathcal{L}_t) $$ is the representation defining $\mathcal{L}$, then $f^* \mathcal{L}$ is described by the composition $$ \pi_1(X) \xrightarrow{f_*} \pi_1(Y) \to \mathrm{Aut}(\mathcal{L}_t). $$

The cohomology groups of a local system on $S^1$ are also nicely described in terms of representations: if we denote by $T \in \mathrm{Aut}(\mathcal{L}_t)$ the image of a generator of $\pi_1 (S^1)$, then $$ H^0(S^1, \mathcal{L}) \cong \mathrm{Ker}(T - id) $$ $$ H^1(S^1, \mathcal{L}) \cong \mathrm{Coker}(T - id) $$ $$ H^n(S^1, \mathcal{L}) = 0), \quad \text{for any } n>1 $$ This is proved using Cech cohomology.

Now the homotopy type of our map $f: S^1 \to S^1$ is determined by its degree $d$, and the automorphism of the stalk that defines $f^* \mathcal{L}$ is just $T^d$. Since $\mathrm{Ker}(T - id) \subset \mathrm{Ker}(T^d - id)$ it seems tempting to say that the pullback map on $H^0$ is the inclusion. However I don't know how to prove this, because I haven't found any result about some possible functoriality of Cech cohomology with respect to the covering. Furthermore, I don't know what the map on $H^1$ could look like. The first candidate I can think of would be some projection between the cokernels, but $\mathrm{Im}(T - id) \subset \mathrm{Im}(T^d - id)$ seems unlikely because of the first isomorphism theorem (if the kernels get larger then the images should get smaller).