  Cobordism groups and cut-and-paste groups of manifolds arise from imposing
two different relations on the monoid of manifolds under disjoint union. By
imposing both relations simultaneously, a cobordism cut and paste group
$\overline{\mathrm{SK}}_n$ is defined. In this paper, we extend this definition
to manifolds with boundary obtaining $\overline{\mathrm{SK}}^{\partial}_n$ and
study the relationship of this group to an appropriately defined cobordism
group of manifolds with boundary. The main results are the construction of a
spectrum that recovers on $\pi_0$ the cobordism cut and paste groups of
manifolds with boundary, $\overline{\mathrm{SK}}^{\partial}_n$, and a map of
spectra that lifts the canonical quotient map $\mathrm{SK}^{\partial}_n
\rightarrow \overline{\mathrm{SK}}^{\partial}_n$.
