  In this paper, we study a generalization of twisted (groupoid) equivariant
$\mathrm{K}$-theory in the sense of Freed-Moore for $\mathbb{Z}_2$-graded
$\mathrm{C}^*$-algebras. It is defined by using Fredholm operators on Hilbert
modules with twisted representations. We compare it with another description
using odd symmetries, which is a generalization of van Daele's
$\mathrm{K}$-theory for $\mathbb{Z}_2$-graded Banach algebras. In particular,
we obtain a simple presentation of the twisted equivariant $\mathrm{K}$-group
when the $\mathrm{C}^*$-algebra is trivially graded. It is applied for the
bulk-edge correspondence of topological insulators with CT-type symmetries.
