For a nonorientable closed surface standardly embedded in the –sphere, a diffeomorphism over this surface is extendable if and only if this diffeomorphism preserves the Guillou–Marin quadratic form of this embedded surface.
"On diffeomorphisms over nonorientable surfaces standardly embedded in the $4$–sphere." Algebr. Geom. Topol. 12 (1) 109 - 130, 2012. https://doi.org/10.2140/agt.2012.12.109