  A decomposition theorem for the Lind zeta function of a reversal system $(X,
T, R)$ of finite order is established. A reversal system can be regarded as an
action of a certain group $G$ on $X$. To establish an explicit formula for the
Lind zeta function of $(X, T, R)$, we need to consider finite index subgroups
$H$ of $G$ with induced actions given by automorphisms or by flips. When the
underlying dynamical system $(X, T)$ is either a shift of finite type or a
sofic shift, we express the Lind zeta function of $(X, T, R)$ in terms of
matrices.
