Whatever be the charge and field configuration outside, any cavity in a conductor remains shielded from outside electric influence. The field inside a conductor is zero. This is known as electrostatic shielding.
Sensitive instruments are shielded from outside electrical influences by enclosing them in a hollow conductor.
During lightning it is safest to sit inside a car, rather than near a tree. The metallic body of a car becomes an electrostatic shielding from lightening.
Potential inside the cavity is not zero. Potential is constant.