Linking in Cyclic Branched Covers and Satellite (non)-Homomorphisms

Let $K\subset S^3$ be a knot and $\eta, \gamma \subset S^3\backslash K$ be simple closed curves. Denote by $\Sigma_q(K)$ the $q$-fold cyclic branched cover of $K$. We give an explicit formula for computing the linking numbers between lifts of $\eta$ and $\gamma$ to $\Sigma_q(K)$. As an application, we evaluate, in a variety of cases, an obstruction to satellite operations inducing homomorphisms on smooth concordance.