In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:

By applying this result to finite groups, we get a Galois correspondence between subgroups and subsystems (of unitary representations).

Such a result seems "fundamental" in the finite group theory but I don't know an "old" reference for it.

Question: What's the first reference for this result in finite group theory?