  KMS states on $\mathbb{Z}_2$-crossed products of unital $C^*$-algebras
$\mathcal{A}$ are characterized in terms of KMS states and twisted KMS
functionals of $\mathcal{A}$. These functionals are shown to describe the
extensions of KMS states $\omega$ on $\mathcal{A}$ to the crossed product
$\mathcal{A} \rtimes \mathbb{Z}_2$ and can also be characterized by the twisted
center of the von Neumann algebra generated by the GNS representation
corresponding to $\omega$.
  As a particular class of examples, KMS states on $\mathbb{Z}_2$-crossed
products of CAR algebras with dynamics and grading given by Bogoliubov
automorphisms are analyzed in detail. In this case, one or two extremal KMS
states are found depending on a Gibbs type condition involving the odd part of
the absolute value of the Hamiltonian.
  As an application in mathematical physics, the extended field algebra of the
Ising QFT is shown to be a $\mathbb{Z}_2$-crossed product of a CAR algebra
which has a unique KMS state.
