SOP's will always vary between differing airlines. Hopefully the one that your company chooses to employ will result in the safe stopping of the aircraft on the runway.
My present employer (747 classic) has the Captain advance the thrust levers towards take-off setting assisted by Flight Engineer.
The Captain calls for, and carries out, any reject that is required. In my company the call of 'STOP' not only means stop but also, 'I HAVE CONTROL' so there is no doubt who is going to perform the RTO and it's associated actions.
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