headman vs public executioner

headman and public executioner both are nouns.

Word	Noun	Adjective	Verb	Adverb
headman	Yes	No	No	No
public executioner	Yes	No	No	No

As nouns, public executioner is a hypernym of headman; that is, public executioner is a word with a broader meaning than headman:

headman: an executioner who beheads the condemned person
public executioner: an official who inflicts capital punishment in pursuit of a warrant

Other hypernyms of headman include executioner.

headman (noun)	public executioner (noun)
the head of a tribe or clan	an official who inflicts capital punishment in pursuit of a warrant
an executioner who beheads the condemned person