F \cong \pi_1(\Sigma)$; the inclusion implies $f^k$ lifts to a map $\hat f: \hat \Sigma \to \hat \Sigma$. Furthermore, up to homotopy, $\hat f$ preserves the core of $\hat \Sigma$, a compact subsurface that supports $A_j$. Let $\gamma \subset \hat \Sigma$ be a peripheral simple closed curve of the core. After replacing $f$ with a power, we can assume $\gamma$ is an $f$-invariant simple closed curve. However, the projection of $\gamma$ to $\Sigma$ may not be a simple closed curve. Using the [*LERF*]{} property of $F \cong \pi_1(\Sigma)$, construct a finite cover $\bar \Sigma$ such that the projection of $\gamma$ to $\Sigma$ lifts to a simple closed curve $\bar \gamma$ in $\bar \Sigma$.

Since $\phi$ is an automorphism, after passing to a power, we can assume $f$ lifts to a homeomorphism $\bar f: \bar \Sigma \to \bar \Sigma$. This map is pseudo-Anosov since the $f$-invariant measured foliations on $\Sigma$ lift to $\bar f$-invariant measured foliations on $\bar \Sigma$. After passing to power again, we can assume $\bar \gamma$ is $\bar f$-invariant. But the only invariant nontrivial simple closed curves of a pseudo-Anosov homemorphism are the peripheral curves. Thus, the projection of $\gamma$ to $\Sigma$ is a (power of a) peripheral 