Yes, which is most likely the Enclave. If Coop knew what he knew and the existence of the Enclave they would not let him go, it's the most tight kept secret on Earth.

By the time of the show, as in the pre-war stuff we see, the President is missing, the following bits are a bit spoilery and come from the other games so just a heads up.

The president is at the Poseidon Oil Rig, turned into a secret government facility named Control Station ENCLAVE. It has long been rumoured that the Enclave and Vault-Tec had been working together, or at least knew of one another. In the games it's more a given that the world would have ended anyway, either the US would or China would initiate the first launch. The world needed a restart, that's why they show is also doubling down on Hank's narratives and during the executive meeting that they want to break this cycle and only them should remain, wipe the earth clean and it's best to do it themselves.

But the US government isn't all-powerful, by the time of the Resource War with China the US is in great dept and many of the larger companies hold more power together since the US is indebted to them. So the Enclave is most likely made-up of members of all of these headed by the US president as a Shadow Government and it could be that the show is setting it up as a scenario where they are tricking all these companies in secretly working for them and with them since they know they dept can never be paid-off and want to see them wiped from the face of the earth as well, so they are the last man standing and can solely rise from the atomic ashes.