  Classical results of Rohlin, Dold, Wall and Atiyah yield two exact sequences
that connect the oriented and unoriented (abstract) cobordism groups $\Omega_n$
and $\mathfrak{N}_n$. In this paper we present analogous exact sequences
connecting the oriented and unoriented cobordism groups of maps with prescribed
singularities. This gives positive answer to a fifteen-year-old question posed
by Sz\H{u}cs and has interesting consequences even in the case of cobordisms of
immersions.
