The question should be clear enough from the title.
Also: What are we supposed to call one who ousts? [If this warrants another question, I will edit this out and open another question.]
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An ouster (noun) is an ejection from an office or a position. Etymonline gives its derivation thus:
oust early 15c., from Anglo-Fr. oster (late 13c.), O.Fr. oster "put out, keep off, remove, avert" (Fr. ôter), from L. obstare "stand opposite to, block, hinder," from ob "against" + stare "to stand," from PIE base sta - "to stand" (see stet).
So the noun derives from the Anglo-French meaning, first and foremost in the sense of a "putting out" of someone, and it has come down to us as a handy synonym of "expulsion" or "impeachment" with the more general sense of relieving officials of their positions.
That said, ouster could be used in both senses: As someone who ousts someone else from a position, and the act of ousting that person. But the use in the former sense would be uncommon and not readily understood with no supporting context.
I think a "courteous" foreign language (French) was used as a veil, because the action is potentially embarrassing; and the infinitive was used as a noun to maintain the foreignness. That has since been forgotten, and the pronunciation Anglicized. I think its recent importation into British English (e.g. in BBC news reports) is willfully Hitchcockian, because it suggests some point is being made about confusing the deed with the doer.
As I understand it, it comes from the French "ouster", where the "-er" suffix indicates that it's a verb.
Manger (pronounced 'mawn-JAY' - "eating")
Parler (pronounced 'par-LAY' - "talking")
Ouster (pronounced 'oos-TAY' - [no equivalent word] [use theirs] [keep the spelling] [give it a new pronunciation] [devise 'oust' as a backformation])