On the number of fixed points of sofic flip systems

In the case when $X$ is a sofic shift and $\phi : X \to X$ is a homeomorphism such that $\phi^2 = \text{id}_X$ and $\phi \sigma_X = \sigma_X^{-1} \phi$, the number of points in $X$ that are fixed by $\sigma_X^m$ and $\sigma_X^n \phi$, $m=1,2,...$, $n\in\Bbb Z$, is expressed in terms of a finite number of square matrices: The matrices are obtained from Krieger's joint state chain of a sofic shift which is conjugate to $X$.