Let $M$ be a non-orientable smooth $d$-dimensional manifold. Let $\tilde{M}$ be the oriented double cover for $M$; and let $f\in Diff^{1+\beta}(M)$. Define $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ the following way: $\tilde{f}((x,o_x)):=(f(x),o(d_xf(b)))$ where $b$ is some basis for $T_xM$ from the equivalence class $o_x$, $o(b')$ is the equivalence class of a basis $b'$.
My question is, how can one show that also $\tilde{f}\in Diff^{1+\beta}(\tilde{M})$ ?
Thanks ahead for any heplers