There is a homomorphism $$\varphi : \{\pm 1\} \longrightarrow \mathrm{Aut}(SL_2(\mathbb{R}))$$ defined by $\varphi(-1)(A) = (A^{-1})^T$ and this allows us to construct the semidirect product $\{\pm 1\} \rtimes_{\varphi} SL_2(\mathbb{R})$.
Since $SL_2(\mathbb{R})$ has homotopy group $\mathbb{Z}$, there should be only one double cover up to isomorphism. I have already been introduced to the metaplectic group $Mp_2(\mathbb{R})$ whose elements are pairs $(A, \varepsilon)$, where $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbb{R})$ and $\varepsilon : \mathbb{H} \rightarrow \mathbb{C}$ is a holomorphic function on the complex upper half-plane such that $\varepsilon(z)^2 = cz+d.$
It isn't clear to me that these are the same group and I think I have probably misunderstood something. I don't think these are isomorphic since an isomorphism would lead to a "natural" choice of square root $\varepsilon$ of $cz+d$. I would be grateful if someone can clarify this.