  In this article we construct examples of non-smoothable
$\mathbb{Z}/p$-actions on indefinite spin 4-manifolds with boundary for all
primes $p\geq 5$. For example, we show that for each prime $p\geq 5$ and each
$n\geq 1$ there exists a locally linear $\mathbb{Z}/p$-action on the Gompf
nucleus $N(2pn)$ which is not smoothable with respect to any smooth structure
on $N(2pn)$. Furthermore we investigate the behavior of these actions under two
different types of equivariant stabilizations with $S^{2}\times S^{2}$, namely
\emph{free} and \emph{homologically trivial} stabilizations -- in particular we
show that our non-smoothable $\mathbb{Z}/p$-action on $N(2pn)$ remains
non-smoothable after $2n-2$ free stabilizations, and after arbitrarily many
homologically trivial stabilizations. We also show that free stabilizations
satisfy a Wall stabilization principle in the sense that any non-smoothable
$\mathbb{Z}/p$-action becomes smoothable after some finite number free
stabilizations (under certain assumptions), whereas our aforementioned result
implies that homologically trivial stabilizations do not satisfy this property.
The proofs of these results use equivariant $\kappa$-invariants defined by the
author in \cite{Mon22}, calculations of equivariant $\eta$-invariants for the
odd signature and Dirac operators on Seifert-fibered spaces, as well as an
analysis of the geometric $S^{1}$-action on the Seiberg-Witten moduli spaces of
Seifert-fibered spaces induced by rotation in the fibers, which may be of
independent interest.
