The eta function and eta invariant of $\mathbb{Z}_{2^r}$-manifolds

We compute the eta function $\eta(s)$ and its corresponding $\eta$-invariant for the Atiyah-Patodi-Singer operator $\mathcal{D}$ acting on an orientable compact flat manifold of dimension $n =4h-1$, $h\ge 1$, and holonomy group $F\simeq \mathbb{Z}_{2^r}$, $r\in \mathbb{N}$. We show that $\eta(s)$ is a simple entire function times $L(s,\chi_4)$, the $L$-function associated to the primitive Dirichlet character modulo 4. The $\eta$-invariant is 0 or equals $\pm 2^k$ for some $k\ge 0$ depending on $r$ and $n$. Furthermore, we construct an infinite family $\mathcal{F}$ of orientable $\mathbb{Z}_{2^r}$-manifolds with $F\subset \mathrm{SO}(n,\mathbb{Z})$. For the manifolds $M\in \mathcal{F}$ we have $\eta(M)=-\tfrac{1}{2}|T|$, where $T$ is the torsion subgroup of $H_1(M,\mathbb{Z})$, and that $\eta(M)$ determines the whole eta function $\eta(s,M)$.