Suppose $f \colon \mathrm{Spec}\, L \to \mathrm{Spec}\, K$ be a finite covering. Given a smooth sheaf ${\cal F}_{\rho}$ on $\mathrm{Spec}\,L$ corresponding to the representation $\rho \colon {\pi}_1(\mathrm{Spec}\,L) \to \mathrm{GL}_n(\Bbb{Z}_l)$, we will consider the direct image $f_* {\cal F}_{\rho}$ on $\mathrm{Spec}\,K$.
Q. Why does $f_* {\cal F}_{\rho}$ coincide with the one induced by the induced representation of $\pi_1(\mathrm{Spec}\,K)$ by way of $\pi_1(\mathrm{Spec}\,L) \to \pi_1(\mathrm{Spec}\,K)$?