SOPs where I work: HP on TLs < V1, either pilot can call "STOP". HP closes TLs, monitors/overrides autobrakes; NHP selects reverse (and speedbrake if required). PIC takes control (if not HP) sometime between initial actions complete and aircraft coming to a halt.
I hate to be critical but what an astonishing SOP with hands all over the place and a recipe for confusion and potential for a big f@@k up. Why would you override the RTO selection of autobrakes when it already gives you max braking. Fair enough if the auto-brake disarm caution light comes on without any obvious reason - in which case the first officer is supposed to see that and call.
But for the first officer as PF to make the command decision to reject the take off AND leave the reverse thrust to the captain is completely at odds with the Boeing recommendation. On a slippery runway with a crosswind the FCTM advises that reverse might have to be reduced to idle to prevent further yaw into wind then re-applied once the aircraft is straight.
Imagine the frantic mess in the cockpit when the first officer does the take off, calls STOP at high speed because he is allowed to by SOP, leaving the captain open-mouthed in horror and too late to prevent the F/O from taking what may be an irrevocable action. The abort proceeds on the slippery runway with the F/o calling for reverse to go to idle as the 73 goes sideways under the crosswind. Then someone (capt or F/O) disengages the RTO in order to meet the Boeing advice at FCTM page 6.31 which states in part: "to correct back to centreline, reduce reverse thrust to reverse idle and
release the brakes. . When re-established near the runway centreline, apply max braking and symmetrical reverse to stop the airplane.
A rejected take off on a slippery surface and a crosswind takes very careful handling if you follow the Boeing procedure above. Once the aircraft starts to drift sideways at an angle due weather-cocking on a wet/slippery runway then unless immediate action is taken to correct the situation, control of the aborted take off may be lost because things happen so quickly. To have two pilots "sharing" the various controls on an abort is asking for confusion.
The rejected take off decision is the captains responsibility entirely and he should have full control of all the flight deck systems that are required during an abort. That means the brakes, thrust levers, speed brake and reverse thrust levers. He then coordinates his actions on each of the systems to achieve a safe result.
Note: Personal opinion only but backed up by reference to the manufacturers recommendations