Regulators

Fallout
Lead by the sadistic Caleb, the Regulators are a vicious gang that serves as the "police force" of Adytum in Southern California. Nominally under the authority of Adytum's mayor, Jon Zimmerman, they in fact manipulate Zimmerman and control Adytum's affairs, ruling harshly, expropriating a good part of Adytum's industrial wealth for themselves, and enforcing forced labor on Adytum's citizens. To legitimate their authority, they have claimed that the Blades, a survivalist group to the north, are a threat, and frame them for crimes such as the murder of Jon Zimmerman's son.

Fallout 3
It must be noted that Fallout 3's Regulators are NOT affiliated in any way with Caleb's Regulators in Fallout 1. They share absolutely nothing in common other than, out of pure coincidence, the same name.

In the Capital Wasteland, the Regulators are a group of self-appointed lawmen who wear cowboy style hats and dusters after the fashion of gunslingers and operate out of a farmhouse, lead by Sonora Cruz. If the player has the Lawbringer perk, he/she can trade fingers taken from the corpses of evil humans killed by the player, for good karma and Bottlecaps. The Regulators will also attempt to claim the bounty on the player's head if he or she commits serious crimes.

Regulators wear Regulator Dusters, and seem to have a fondness for laser weapons (laser pistols and laser rifles). In terms of durability and combat challenge, they're about equal to a Talon Company Merc. Regulator patrols typically operate in 3-man squads when searching the Wasteland for the player character if that player is evil or very evil.

If you have the Lawbringer perk, and ONLY if you do, a dead Regulator will be found outside of the entance to Arlington/Wasteland Metro in Arlington Cemetery North. He will have a bounty note for Junders Plunkett. It says he was last seen in Canterbury Commons, but really he is in the nearby Arlington House. Kill him and take his finger. It is worth 1,000 caps back at the Regulator HQ.

Bugs
There is a bug where a player can pick-pocket a regulator, and he/she will have a named bounty for the player on them, even if the player is completely good.

Sometimes if the character fast-travels to the Regulator HQ, a giant radscorpion will be attacking the brahmin outside. If the character enters the HQ without killing the scorpion, Sonora Cruz can run out of the HQ into the wasteland and be presumably killed.

This may be bugged, as you can retrieve both a normal finger and a named finger from the same target, then turn both in to Sonora Cruz at Regulator HQ for the normal and special bounties. The named finger remains in your inventory.

Note that the Regulators will target you if your Karma ever goes to "evil". Even if your karma dips to evil (i.e. if you enslave a few Raiders using the Mesmetron), and you quickly get it back up to "good" (i.e. by giving purified water to beggars) without encountering the Regulators, you will eventually encounter hostile Regulators even as a "good" character. Sometimes, a squad of 3 Regulators will even appear alongside a squad of 3 Talon Company Mercs to fight you! Ironically, they usually start fighting each other when you encountered both of them. These Regulators also carry bounty notices against you.

There is a bug, on all three versions, that Sonora Cruz will 'flee' from you if you approach her for no reason regardless of having good Karma. If you DO approach her she will run from the Regulator HQ and then you will see a message "Sonora Cruz is unconscious" (as an NPC she'll be easily picked off by the large animals or Enclave troops in the area). Returning to Regulator HQ after this finds her gone and nothing to do with your growing collection of fingers. It seems that this "bug" happens when you enter the HQ after being spotted by hostile NPC. To prevent this from happenning, "sneak" before entering the HQ, if you are "HIDDEN", you can enter safely, otherwise you will need to take care of the hostiles (probably giant radscorpions)

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Appearances in games
Different groups going by the name "Regulators" appear in the original Fallout and in Fallout 3.