None are likely to play out well for Russia.

In a speech to the nation early last week, Mr. Putin said that Wagner fighters who did not participate in the coup attempt were free to sign a contract with the Russian military under the command of the Ministry of Defense. (Those who did participate could join Mr. Prigozhin in Belarusian exile.) But the mandate for Wagner mercenaries to sign a contract with the Russian Ministry of Defense — a policy Defense Minister Sergei Shoigu tried to put in place in early June — was one of the primary factors contributing to Mr. Prigozhin’s mutiny attempt, and isn’t likely to be a popular choice for his troops.

Even if Wagner fighters do decide to join Russian military units en masse, it won’t be easy for Moscow to integrate them. Wagner forces already have a well-deserved reputation for brutality, are alleged to have committed war crimes and crimes against humanity in several theaters and have been credibly accused of torture, kidnapping and executing civilians.

Another option would be for Mr. Putin to leave Wagner’s overseas operations as is, and install a new leader to replace Mr. Prigozhin. That would avoid disrupting Moscow’s foreign policy agenda and reassure its client states that Russia remains a reliable partner. Wagner’s African footprint is vast, with ongoing activities in Cameroon, Burkina Faso, Equatorial Guinea, South Africa, Zimbabwe and Kenya. When asked immediately after the failed uprising what would happen to Wagner’s presence in Africa, Russia’s foreign minister, Sergei Lavrov, announced that Russian “instructors” would continue working in Mali and the Central African Republic. Russia has since dispatched officials to various spots where Wagner operates to reassure those governments, including Syria and Mali, that there will be no disruption in Russian assistance.

But that, too, could get messy, depending on how deep the rift between Wagner and the Ministry of Defense runs. If Wagner’s midlevel commanders and foot soldiers remain loyal to Mr. Prigozhin, installing a new figurehead with the Kremlin’s imprimatur may not work. Mr. Prigozhin was revered by Wagner fighters, many of whom may chafe at the prospect of new leadership or a drastic change in organizational culture.

Finally, Russia could seek to disband Wagner entirely and disperse its fighters into existing private armies. Patriot, a group linked to Mr. Shoigu, is widely viewed as a significant competitor to Wagner, with reported operations in Burundi, the Central African Republic, Gabon, Syria and Yemen. The E.N.O.T. Corporation is another Russian private military company, founded by Russian nationalist Igor Mangushev, with some experience abroad, but is much less influential and experienced than Wagner. Gazprom, the Russian energy giant, has also developed its own private army, although it is designed mainly to protect oil and gas infrastructure against attacks.