Let $\rho_1$, $\rho_2$ be representations of $SL(2)$. Given arbitrary $f:V \to \mathbb{R}$, is it possible to construct an "averaged" function such that for all $g \in SL(2, \mathbb{R})$, $\tilde{f}(\rho_1(g) x) = \rho_2(g) \tilde{f}(x)$, along the lines of $$ \tilde{f}(x) = \int_{h\in SO(2)} f(\rho_1(h)x) dh?$$
Given the end of the answer to this question, I would infer something like the above is true, where one only needs to integrate over the maximally compact subgroup. However, the computation above does not seem to be correct computationally. What exactly is the Theorem statement of Weyl's unitary trick in this context?