Lifting braids over $\mathbb{RP}^2$ and non-orientable Lefschetz fibrations

We show that, given a closed connected non-orientable surface $F$ and a simple branched cover $p \colon F \rightarrow \mathbb{RP}^2$ of degree $d \geq 5$, every self-diffeomorphism of $F$ is isotopic to the lift of a self-diffeomorphism of $\mathbb{RP}^2$ with respect to $p$. As an application, we s