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Here is a simple counter-example. Let $p : S_3 \to S_2$ be a degree 2 covering map from the closed genus 3 surface to the closed genus 2 surface, inducing an index 2 injection $p_* : \pi_1(S_3) \to \pi_1(S_2)$. Pulling back any hyperbolic structure from $S_2$ to $S_3$ via $p$ defines an embedding of Teichmuller spaces $p^* : T(S_2) \to T(S_3)$, the domain having dimension 6 and the range having dimension 12. Pick a point in $T(S_3)$ which is not in the image of this embedding nor in any translate of the image under the action of the mapping class group $MCG(S_3)$ on $T(S_3)$. This point represents a discrete subgroup $\Gamma \approx \pi_1(S_3)$ of $Isom(H^2) = PSL(2,R)$ such that, under the index 2 inclusion of $\Gamma \hookrightarrow \tilde\Gamma \approx \pi_1(S_2)$, the action of $\Gamma$ does not extend to an action of $\tilde\Gamma$.