Twisted spin bordism invariants in 5 dimensions [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:

*

*

*

*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}$,




*


*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.
$$
\text{the $\eta$-invariant of the Dirac operator acting on
the (twisted) Dirac spinor bundle}
$$
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 $A \cup \eta$?



*


*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
$B' \in H^2(B \mathbb Z_{4}, \mathbb Z_{2})=\mathbb Z_{2}$.



*


*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.



 A: $\newcommand{\Z}{\mathbb Z}\newcommand{\RP}{\mathbb{RP}}$
Let $G_1 := \mathrm{Spin}_5\times_{\Z/2}\Z/4$.  The group $\Omega_5^{G_1}$ is discussed by
Tachikawa-Yonekura, §3.1 and §3.4.  Given a 5-manifold $M$ with a
$G_1$-structure given by the principal $G_1$-bundle $P\to M$, there's a principal $\Z/2$-bundle $Q := P\times_{G_1}
\Z/2\to M$. If $N\subset M$ is a representative of the Poincaré dual to $w_1(Q)\in H^1(M;\Z/2)$, then $N$ acquires
a pin$+$-structure. Then:


*

*The map $\Omega_5^{G_1}\to\Omega_4^{\mathrm{Pin}^+}\cong\Z/16$ sending $M$ to the bordism class of $N$ is an
isomorphism; in particular, one can realize the isomorphism $\Omega_4^{\mathrm{Pin}^+}\to\Z/16$ as an
$\eta$-invariant, and putting this together gives a complete bordism invariant for $\Omega_5^{G_1}$.

*A manifold generator of $\Omega_5^{G_1}$ is $\RP^5$ with a $G_1$-structure such that $Q$ is the nontrivial
principal $\Z/2$-bundle; then an embedded $\RP^4\subset\RP^5$ represents the Poincaré dual of $w_1(Q)$ and
$\RP^4$ acquires a pin$+$-structure representing $1$ or $-1$ in
$\Omega_4^{\mathrm{Pin}^+}\cong\Z/16$.


Let $G_2 := \mathrm{Spin}_5\times_{\Z/2}\Z/8$. Then $\Omega_5^{G_2}$ is discussed by
Hsieh, §2.2. In particular:


*

*Given an element $R$ of the representation ring of $\Z/8$, Hsieh shows how to define an exponentiated
  $\eta$-invariant $\eta_R\colon\Omega_5^{G_2}\to\mathrm U_1$, and proves that these are complete invariants,
  i.e. there is some combination of $\eta$-invariants for some of these representations realizing the maps
  $\Omega_5^{G_2}\to\Z/32$ and $\Omega_5^{G_2}\to\Z/2$ that you're interested in.

*Lens spaces, with various $G_2$-structures, generate $\Omega_5^{G_2}$.


Given $R$ as above and a lens space $L(m, \vec a)$ with $G_2$-structure, Hsieh provides a formula (equation (2.35)
in the paper) for $\eta_R(L(m, \vec a))$; this should make it possible to explicitly determine which
$\eta$-invariants and which lens spaces you're looking for.
