FAR 117.19 states that the decision to extend is made jointly by the certificate holder (the company) and the PIC (the captain). From a technical standpoint, the FO never enters the equation unless he has been previously extended greater than 30 minutes and has not had the requisite 30 consecutive hours free from all duty. If that's the case, the crew cannot be extended.

That being said, any respectable captain is going to gather the input from his crew before making his decision on the extension. I would expect that to be the norm, but the letter of the law does not require it.

Additionally, consider the situation where an FO signed the release prior to pushback but during taxi out the crew is notified by ATC of a ground stop and they are delayed. If the delay leads to a decision point on an extension, the FO has already signed and is in the position described above.