  We prove that the solid ergodicity property is stable with respect to taking
coinduction for a fairly large class of coinduced action. More precisely,
assume that $\Sigma<\Gamma$ are countable groups such that $g\Sigma g^{-1}\cap
\Sigma$ is finite for any $g\in\Gamma\setminus\Sigma$. Then any measure
preserving action $\Sigma\curvearrowright X_0$ gives rise to a solidly ergodic
equivalence relation if and only if the equivalence relation of the associated
coinduced action $\Gamma\curvearrowright X$ is solidly ergodic. We also obtain
orbit equivalence rigidity for such actions by showing that the orbit
equivalence relation of a rigid or compact measure preserving action
$\Sigma\curvearrowright X_0$ of a property (T) group is "remembered" by the
orbit equivalence relation of $\Gamma\curvearrowright X$.
