Biases towards claiming agreement with one’s own beliefs
If the institution is widely trusted, respected, high status, etc., as well as powerful, then if Alice convinces you that the institution supports her beliefs, then you might be inclined to give more credence to Alice’s beliefs. That would serve Alice’s political agenda.
Weaker biases towards claiming disagreement with one’s own beliefs
If the institution is widely hated—for example al-Qaeda, the CIA, the KGB—or considered low status, crazy, and so on, then if Alice convinces you that the institution opposes her beliefs, that might make you more sympathetic to her, make you distrust arguments against her beliefs, and/or defuse preexisting arguments that support for Alice’s position comes mostly from these evil/crazy institutions.
Other reasons:
If the institution is widely trusted, respected, high status, etc., as well as powerful, then if Alice convinces you that the institution supports her beliefs, then you might be inclined to give more credence to Alice’s beliefs. That would serve Alice’s political agenda.
If the institution is widely hated—for example al-Qaeda, the CIA, the KGB—or considered low status, crazy, and so on, then if Alice convinces you that the institution opposes her beliefs, that might make you more sympathetic to her, make you distrust arguments against her beliefs, and/or defuse preexisting arguments that support for Alice’s position comes mostly from these evil/crazy institutions.