We extend the definitions and main properties of graded extensions to the category of locally compact groupoids endowed with in-volutions. We introduce RealČechRealˇRealČech cohomology, which is an equivariant-like cohomology theory suitable for the context of groupoids with invo-lutions. The Picard group of such a groupoid is discussed and is given a cohomological picture. Eventually, we generalize Crainic's result, about the differential cohomology of a proper Lie groupoid with coefficients in a given representation, to the topological case.