Etymology of 'remit' {noun}?

As a native British English speaker, I have a vague preference for stressing the first syllable for #1: "I have a wide REmit", and stressing the second for something related to remittance ("the Court of Session may accept a reMIT").

A memory aid could be:

  1. REmit is like GAmut; this is the stuff encompassed by a job, like a gamut is the colours encompassed by a printer.
  2. reMIT is like a reTURN in tennis, a court can agree to a reMIT and thus a reTURN of the case to a lower court. As "a return" is to "I return", so "A remit" is to "I remit".