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Any interested party can file for probate, meaning, to initiate probate. This may be a creditor, a family member, or a close friend who believes that the Will left them something.
Once a probate matter is initiated, the Court will appoint the Executor (also known as the Administrator, or, Personal Representative). Normally this is the person that the WILL states will fulfill this role, but if this person is unable or unwilling to serve, the Court will appoint someone else. Interested parties can nominate themselves or others, and then the Judge will decide.
Once the Executor is nominated, they have a duty to fulfill the wishes of the deceased as written out in the Will.
I am not sure the Court appointed a Public Administrator. Normally this happens if the Executor does not file for probate, or is contacted but declines to serve. Also, there may have been a misunderstanding that she did not have a Will, because until/unless someone files, no one knows for sure.
If the Court appointed a Public Administrator, then the Public Administrator is the party responsible to carry out the wishes of the Will. If the Court orders a Will to be proven up, then whoever the Court orders to do so must do so.
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