  We study the partially wrapped Fukaya category of a surface with boundary
with an action of a group of order two. Inspired by skew-group algebras and
categories, we define the notion of a skew-group $A_\infty$-category and let it
play the role of the partially wrapped Fukaya category of an orbifold surface.
We classify indecomposable objects in terms of graded curves with signs, or
taggings, at orbifold points. We compute morphisms between a class of objects,
and we use this to describe tilting objects and find algebras
derived-equivalent to skew-gentle algebras.
