This would really depend on the circumstances of the case (most countries aren't going to waste considerable time and expense extraditing you unless it's for something serious, even if you want it) and the country concerned. People extradited to the United States generally won't be issued a visa, but are paroled into the United States, which is essentially a catch-all category for people who the US government wants to admit to the country for all sorts of reasons outside of the usual visa rules. The State Department's Foreign Affairs manual describes this process:
a. All extradited fugitives whose U.S. citizenship cannot be confirmed are processed as aliens and paroled into the United States regardless of any previous resident status.
b. The Parole and Humanitarian Assistance Branch (PHAB), within the Department of Homeland Security (DHS), Immigration and Customs Enforcement (ICE), Office of International Affairs, undertakes parole of an alien fugitive into the United States, in accordance with Section 212(d)(5)(A) of the Immigration and Nationality Act (8 U.S.C. 1182(d)(5)(A)). The Parole facilitates the return of a fugitive as air and other carriers are prohibited from carrying as a passenger an individual who is not properly documented to enter the United States.
c. Because these requests involve timeliness and sensitivity, they are expeditiously processed. An alien whom PHAB paroles into the United States for purposes of trial or service of a sentence is under a detainer while serving sentence in the United States.
If it's necessary to facilitate travel by air, the consulate can issue a boarding letter (a document that instructs an air carrier to allow a passenger to board even if they don't have the travel documents that would usually be required for such a trip).
Since the person concerned would likely be considered a flight risk, they're probably going to be detained while awaiting trial and would be deported once their time with the justice system concludes. There's no concern about overstaying, as the parole status lasts as long as the US government wants.
I suspect most countries will have some sort of exceptional immigration status that boils down to "this person is admitted for the convenience of the government for as long as the government wants them here," as there are always unusual situations that require special treatment.