[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance!

The spin $G$-bordism invariant can be twisted in the way that the Spin($d$) group of the $d$-manifold and the $G$ group can be combined and mod out any shared normal subgroup.

For example, we can consider $Spin(d) \times_N G \equiv \frac{Spin(d) \times G }{N}$.

Here we denote $G_1 \times_N G_2 \equiv \frac{G_1 \times G_2 }{N}$ in general.

I am trying to understand a particular twisted spin bordism invariant in 5 dimensions, based on a computation of Adams spectral sequence.

I have obtained that 5d bordism groups as

$$ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{2}}= 0, $$ $$ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{4}} =\mathbb Z_{16}, $$ $$ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}, $$

My question: What are the above $\mathbb Z_{32}$ generator, $\mathbb Z_{16}$ generator, and $\mathbb Z_{2}$ generator as(1) the 5d topological terms, and

(2) its 5d manifold generators?

**What I have done** which leads to a tentative clue for an answer:

1) I find Adams $\mathcal A$ module structure has the following for $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{4}} =\mathbb Z_{16}$ and $ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}$,

2) The $\mathbb Z_{16}$ in $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{4}} =\mathbb Z_{16}$ is similar to the $$ \Omega_4^{Pin^+}(pt) =\mathbb Z_{16}$$, which is well-known to be generated by a 4d $\eta$-invariant that can be obtained from a Dirac spinor in 4d. $$ \text{the $\eta$-invariant of the Dirac operator acting on the 8twisted) Dirac spinor bundle} $$ Thus $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{4}} =\mathbb Z_{16} $ may be generated by $$A \cup \eta?$$ except that there is only an $A \in H^1(B \mathbb Z_2,\mathbb Z_2)=\mathbb Z_2$, there is no ${\mathbb Z_{16}}$ class of $A \cup \eta$?

3) The $\mathbb Z_{2}$ in $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}$ may be generated by $$ A' B' \cup \text{Arf} $$ where $A' \in H^1(B \mathbb Z_{4}, \mathbb Z_{2})=\mathbb Z_{2}$ and $B' \in H^2(B \mathbb Z_{4}, \mathbb Z_{2})=\mathbb Z_{2}$.

4) The $\mathbb Z_{32}$ in $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}$ may be related to the earlier $\mathbb Z_{16}$ and has something to do with the Postnikov square.