Given a free group on $n$ generators, $F_n$, $\phi$ an automorphism of $F_n$, and a non-trivial representation $\rho: F_n \rightarrow Homeo_+(\mathbb{R})$, are necessary and sufficient conditions known, in terms of $\rho, \phi$, and $n$, for extending the representation $\rho$ to $\tilde{\rho}: F_n \rtimes_\phi \mathbb{Z} \rightarrow Homeo_+(\mathbb{R})$?

Given a free group on $n$ generators, $F_n$, $\phi$ an automorphism of $F_n$, and a non-trivial representation $\rho: F_n \rightarrow \operatorname{Homeo}_+(\mathbb{R})$, are necessary and sufficient conditions known, in terms of $\rho, \phi$, and $n$, for extending the representation $\rho$ to $\tilde{\rho}: F_n \rtimes_\phi \mathbb{Z} \rightarrow \operatorname{Homeo}_+(\mathbb{R})$?