  Given a spin rational homology sphere $Y$ equipped with a
$\mathbb{Z}/m$-action preserving the spin structure, we use the Seiberg--Witten
equations to define equivariant refinements of the invariant $\kappa(Y)$ from
\cite{Man14}, which take the form of a finite subset of elements in a lattice
constructed from the representation ring of a twisted product of
$\text{Pin}(2)$ and $\mathbb{Z}/m$. The main theorems consist of equivariant
relative 10/8-ths type inequalities for spin equivariant cobordisms between
rational homology spheres. We provide applications to knot concordance, give
obstructions to extending cyclic group actions to spin fillings, and via taking
branched covers we obtain genus bounds for knots in punctured 4-manifolds. In
some cases, these bounds are strong enough to determine the relative genus for
a large class of knots within certain homology classes in $\mathbb{C}
P^{2}\#\mathbb{C} P^{2}$, $S^{2}\times S^{2}\# S^{2}\times S^{2}$, $\mathbb{C}
P^{2}\# S^{2}\times S^{2}$, and homotopy $K3$ surfaces.
