Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group Let $M^d$ be a nontrivial bordism generator for the bordism group
$$
\Omega_d^G= \mathbb{Z}_n,
$$
suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}_n$ is a finite group. The $M^d$ is the bordism manifold generator genearting the bordism group $\Omega_d^G= \mathbb{Z}_n$.
Suppose some integral cohomology class of $M^d$ is an integral class
$$
H^k(M^d, \mathbb{Z})=\mathbb{Z}.
$$
I suspect that there exists NO extended group $G'$, 


*

*Such that $$
G' \to G
$$
is surjective by any group extension of $N$ (so $G=G'/N$ is a quotient group of $G'$)

*The bordism invariant $M^d$ which is a nonzero generator of $\Omega_d^G$, but is a zero element (trivial element) in
$$
\Omega_d^{G'}.
$$

My question: Is it true about my statement on None existences of any extended group $G'$ satisfy all the criteria 1 and 2 above? How can we prove it rigorously (or at least based on whatever claim we can convince the statement is true?) Or at least is there some argument for my statement to be true, or a counter example for  my statement to be false?

 A: Your suspicion is correct: such a $G'$ cannot exist. The idea is that a $G'$-structure determines a $G$-structure,
so if $M = \partial W$ as $G'$-manifolds, then $M = \partial W$ as $G$-manifolds.

In a bit more detail, suppose $\mathfrak U = \{(U, \phi_U\colon\mathbb R^n\to U)\}$ is an atlas for $M$ such that
$TM|_U\cong U\times\mathbb R^n$ for all charts $U$. If $U$ and $V$ are intersecting charts, there are transition functions $g_{UV} := d\phi_V^{-1}\circ d\phi_U$ from the description of $TM$ in $U$-coordinates to the description
in $V$-coordinates; they are invertible, so define maps $g_{UV}\colon U\cap V\to\mathrm{GL}_n(\mathbb R)$.
Moreover, the transition functions satisfy a cocycle condition on triple intersections $U\cap V\cap W$:
$$ g_{WU}\circ g_{VW}\circ g_{UV} = 1.$$
Now suppose $\rho\colon G\to\mathrm{GL}_n(\mathbb R)$ is a homomorphism of topological groups. There are a few ways
to define a $G$-structure on $M$; one hands-on definition is a lift of the transition functions across $\rho$. That
is, a $G$-structure is data of functions $\widetilde g_{UV}\colon U\cap V\to G$ for all $U,V\in\mathfrak U$ such
that $\rho\circ\widetilde g_{UV} = g_{UV}$ and the cocycle condition is satisfied: $\widetilde
g_{WU}\circ\widetilde g_{VW}\circ \widetilde g_{UV} = 1$ on all triple intersections. (We also have to specify that
$\widetilde g_{UU} = 1$.)
For example, a Riemannian metric determines an $\mathrm O_n$-structure; then an orientation is an
$\mathrm{SO}_n$-structure, and a spin structure is a $\mathrm{Spin}_n$-structure.
The point of all this is that if $\psi\colon G'\to G$ is a homomorphism of topological groups, $\rho\circ\psi\colon
G'\to\mathrm{GL}_n(\mathbb R)$ allows us to define $G'$-structures on manifolds. The key fact is that a
$G'$-structure $\{\widetilde g_{UV}\}$ determines a $G$-structure $\{\psi\circ\widetilde g_{UV}\}$, and this is
compatible with taking boundaries (a $G$-structure on $W$ induces a $G$-structure on $\partial W$ by restricting
the transition maps to the boundary). If
$[M] = 0$ in $\Omega_d^{G'}$, there is a compact $(d+1)$-dimensional $G'$-manifold $W$ such that $M = \partial W$
as $G'$-manifolds. Composing with $\psi$, we obtain $G$-structures on $M$ and $W$, and see that $M = \partial W$ as
$G$-manifolds, which contradicts the initial hypothesis that $[M]\ne 0$ in $\Omega_d^G$.
