Boundary and lens rigidity of finite quotients

We consider compact Riemannian manifolds $(M,\partial M,g)$ with boundary $\partial M$ and metric $g$ on which a finite group $\Gamma$ acts freely. We determine the extent to which certain rigidity properties of $(M,\partial M,g)$ descend to the quotient $(M/\Gamma,\partial/\Gamma,g)$. In particular, we show by example that if $(M,\partial M,g)$ is boundary rigid, then $(M/\Gamma,\partial/\Gamma,g)$ need not be. On the other hand, lens rigidity of $(M,\partial M,g)$ does pass to the quotient.