  Let $Y\to X$ be a finite normal cover of a wedge of $n\geq 3$ circles. We
prove that for any $v\neq 0\in H_1(Y;\mathbb{Q})$ there exists a lift
$\widetilde{F}$ to $Y$ of a homotopy equivalence $F:X\to X$ so that the set of
iterates $\{\widetilde{F}^d(v): d\in \mathbb{Z}\}\subseteq H_1(Y;\mathbb{Q})$
is infinite. The main achievement of this paper is the use of representation
theory to prove the existence of a purely topological object that seems to be
inaccessible via topology.
