In Shimura's Introduction to Arithmetic Theory of Automorphic Functions, the following is given:
Proposition 2.4: Let $\Gamma,\Gamma'<SL_2(\mathbb{R})$ be Fuchsian and $\alpha\in GL_2^+(\mathbb{R})$ such that $\alpha\Gamma\alpha^{-1}<\Gamma'$ with finite index. Then $f\mapsto f\mid[\alpha]_k$ gives a $\mathbb{C}$-linear injections $A_k(\Gamma')\hookrightarrow A_k(\Gamma)$, $G_k(\Gamma')\hookrightarrow G_k(\Gamma)$, and $S_k(\Gamma')\hookrightarrow S_k(\Gamma)$, which are surjective if $\Gamma'=\alpha\Gamma\alpha^{-1}$.
Proof: Let $C$ (resp. $C'$) denote the set of cusps of $\Gamma$ (resp. $\Gamma'$). Then $\alpha(C)=C'$ and our assertion follows immediately from the definition.
The containment $\alpha(C)\subset C'$ is relatively straightforward to show, but where does the reverse containment $\alpha(C)\supset C'$ come from without using the equality at the end of the Proposition statement? (Or does the finite index have something to do with this?)
Additionally, it seems that the non-surjection case can be proved without the reverse containment. Is it the case that that containment is only needed for the surjections?
Thanks in advance for any answers.
The cusps of $\Gamma'\backslash\Bbb{H}$ are the same as those of $\alpha \Gamma \alpha^{-1}\backslash\Bbb{H}$, except that the orbits under the action of $\Gamma'$ on those cusps are larger than the orbits under the action of $\alpha \Gamma \alpha^{-1}$.
The cusps of $\Gamma\backslash\Bbb{H}$ correspond to those of $\alpha \Gamma \alpha^{-1}\backslash\Bbb{H}$ through $z\to \alpha(z)$.