From Latin 'exsequi' (to follow out) — to follow a command through to completion. The death sense: following a sentence to its end.
To carry out or put into effect a plan, order, or course of action; to perform an activity; to put to death as a legal punishment.
From Old French 'executer,' from Medieval Latin 'executāre,' from Latin 'exsecūtus,' past participle of 'exsequī' (to follow out, to pursue to the end, to carry a thing through from beginning to completion, to perform, to enforce, to avenge). The compound is 'ex-' (out, denoting completion and thoroughness) + 'sequī' (to follow), from Proto-Indo-European *sekw- (to follow, to go after). The root *sekw- is one of the most productive in all of Indo-European: Latin 'sequī' (to follow — root of 'sequence,' 'consequence,' 'second' — the one who follows first, 'sect' — a following or faction, 'suit' — a following of law, 'pursue' — to follow after, 'sue' — to follow in law), 'socius' (companion — one who follows along → 'society,' 'social,' 'associate'), 'signum' (a sign, a following-mark — perhaps related), Greek 'hepesthai' (to follow), Sanskrit 'sacate' (accompanies, follows — related to 'saha,' together with), and Old English 'secgan' (perhaps distantly
The word 'executive' — a person who carries out plans and decisions — comes from the same root. The executive branch of government is the branch that follows through on (executes) the laws made by the legislature. 'Executor' of a will is someone who follows through on a deceased person's wishes. All three roles — executive, executor, executioner — are people who follow things through to completion