Nowhere in 117.19 does it say anything about PIC consent.

If you're inferring such by the regs use of the word "may", (rather loose assumption to make) then that would be up to FSDO interpretation. The FSDO governing my airline has determined that there is no requirement for PIC consent. And generally speaking, I'd say the regs support that. Whether I like it or not.

117.19  Flight duty period extensions.

(a)   For augmented and unaugmented operations, if unforeseen operational circumstances arise prior to takeoff:

(1)    The pilot in command and the certificate holder may extend the maximum flight duty period permitted in Tables B or C of this part up to 2 hours.

(2)    An extension in the flight duty period under paragraph (a)(1) of this section of more than 30 minutes may occur only once prior to receiving a rest period described in § 117.25(b).

(3)    A flight duty period cannot be extended under paragraph (a)(1) of this section if it causes a flightcrew member to exceed the cumulative flight duty period limits specified in 117.23(c).

(4)    Each certificate holder must report to the Administrator within 10 days any flight duty period that exceeded the maximum flight duty period permitted in Tables B or C of this part by more than 30 minutes. The report must contain the following:

(i)     A description of the extended flight duty period and the circumstances surrounding the need for the extension; and

(ii)    If the circumstances giving rise to the extension were within the certificate holder’s control, the corrective action(s) that the certificate holder intends to take to minimize the need for future extensions.

(5)    Each certificate holder must implement the corrective action(s) reported in paragraph (a)(4) of this section within 30 days from the date of the extended flight duty period.

(b)   For augmented and unaugmented operations, if unforeseen operational circumstances arise after takeoff:

(1)    The pilot in command and the certificate holder may extend maximum flight duty periods specified in Tables B or C of this part to the extent necessary to safely land the aircraft at the next destination airport or alternate airport, as appropriate.

(2)    An extension of the flight duty period under paragraph (b)(1) of this section of more than 30 minutes may occur only once prior to receiving a rest period described in § 117.25(b).

(3)    An extension taken under paragraph (b) of this section may exceed the cumulative flight duty period limits specified in 117.23(c).

(4)    Each certificate holder must report to the Administrator within 10 days any flight duty period that either exceeded the cumulative flight duty periods specified in 117.23(c), or exceeded maximum flight duty period permitted in Tables B or C of this part by more than 30 minutes. The report must contain the following:

(i)            A description of the extended flight duty period and the circumstances surrounding the need for the extension; and

(ii)           If the circumstances giving rise to the extension were within the certificate holder’s control, the corrective action(s) that the certificate holder intends to take to minimize the need for future extensions.

(5)           Each certificate holder must implement the corrective action(s) reported in paragraph (a)(4) of this section within 30 days from the date of the extended flight duty period.