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.