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 $\tilde{\rho}: F_n \rtimes_\phi \mathbb{Z} \rightarrow \operatorname{Homeo}_+(\mathbb{R})$?
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$\begingroup$ A first necessary condition is that for every $g\in F_n$, the maps $\rho(g)$ and $\rho(\phi(g))$ must be conjugate in $\mathrm{Homeo}_+(\mathbb{R})$. (clearly what you ask is the stronger condition that the conjugating element can be chosen so that it is always the same). Note that this is sufficient in the case $n=1$. $\endgroup$– user47274Commented Jul 31, 2020 at 20:52
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