Knowing the existence of a fixed point set from an induced fundamental group automorphism

Question

Let $L$ be a link in $S^{3} $ and $f_{ \phi } : \pi_{1} (S^{3} \backslash L ) \rightarrow \pi_{1} (S^{3} \backslash L )$ be induced from a periodic map $\phi $ of $S^{3} $, restricted to the complement, with $\phi (L ) = L $. Can I know if $\phi $ has a fixed point set disjoint of $L$ by knowing $f_{ \phi }$?