On Frobenius liftability of surface singularities

We show that a plt surface singularity (P∈ X,B) is F-liftable if and only if it is F-pure and is not a rational double point of type E₈¹ in characteristic $p=5$. As a consequence, we prove the logarithmic extension theorem for F-pure surface pairs and Bogomolov-Sommese vanishing for globally F-split surface pairs. These results were previously known to hold in characteristic $p>5$.