What authority does a pilot-in-command have in an emergency?

6-1-1. Pilot Responsibility and Authority
a. The pilot-in-command of an aircraft is directly responsible for and is the final authority as to the operation
of that aircraft. In an emergency requiring immediate action, the pilot-in-command may deviate from any rule
in 14 CFR part 91, Subpart A, General, and Subpart B, Flight Rules, to the extent required to meet that emergency.
NOTE-
In the event of a pilot incapacitation, an Emergency Autoland system or an emergency descent system may assume operation
of the aircraft and deviate to meet that emergency.
REFERENCE-
14 CFR Section 91.3(b).
b. If the emergency authority of 14 CFR section 91.3(b) is used to deviate from the provisions of an ATC
clearance, the pilot-in-command must notify ATC as soon as possible and obtain an amended clearance.
c. Unless deviation is necessary under the emergency authority of 14 CFR section 91.3, pilots of IFR flights
experiencing two-way radio communications failure are expected to adhere to the procedures prescribed under
"IFR operations, two-way radio communications failure."
REFERENCE-
14 CFR Section 91.185.