  Motivated by Popa's seminal work \cite{Po04}, in this paper, we provide a
fairly large class of examples of group actions $\Gamma \curvearrowright X$
satisfying the extended Neshveyev-St{\o}rmer rigidity phenomenon \cite{NS03}:
whenever $\Lambda \curvearrowright Y$ is a free ergodic pmp action and there is
a $\ast$-isomorphism $\Theta:L^\infty(X)\rtimes \Gamma \rightarrow
L^\infty(Y)\rtimes \Lambda$ such that $\Theta(L(\Gamma))=L(\Lambda)$ then the
actions $\Gamma\curvearrowright X$ and $\Lambda \curvearrowright Y$ are
conjugate (in a way compatible with $\Theta$). We also obtain a complete
description of the intermediate subalgebras of all (possibly non-free) compact
extensions of group actions in the same spirit as the recent results of Suzuki
\cite{Suzuki}. This yields new consequences to the study of rigidity for
crossed product von Neumann algebras and to the classification of subfactors of
finite Jones index.
