New Zealand and England will for the first time compete for the Crowe-Thorpe trophy, which was named in honour of former New Zealand captain Martin Crowe and England batsman Graham Thorpe.
An injury to wicketkeeper Jordan Cox means Ollie Pope will take the gloves and drop down the order to bat at six with Bethell coming in ahead of Joe Root, who will play his 150th Test.
While England are already out of the running, New Zealand could qualify for next year’s World Test Championship (WTC) final with a clean sweep of the series.