Extermination is a Legendary medal awarded in Halo 3, Halo: Reach and Halo 4 multiplayer for killing the last alive player on the opposing team with at least an Overkill that results in the entire team being dead at once. In the first two games it is represented by a gray cross with a red skull. In Halo 4 it represented by a white skull on a red hexagon on top of a 16-pointed grey star.
This medal is extremely difficult to earn, as it requires an entire team to be in the same area, and it is hard to stay alive while killing at least four enemies within four seconds of each other .
It is helpful to note that just getting an overkill on a four-on-four match in Matchmaking will not necessarily achieve the medal. This instance doesn't occur often. In order to unlock the medal, a player must kill the entire team so fast that every member of the team is simultaneously dead for at least a moment. If any respawn before you get the overkill, you will not get the medal.
Perhaps the hardest conventional team slayer playlist to get this medal on would be Slayer, as it has a battle consisting of a four-on-four match. Excluding this, achieving this medal on Social/Mythic Big Team Battle would be extremely noteworthy, because that would either require getting a Killtastrophe on the enemy team before any of them respawn, or significant teamwork and luck.
A fact about the Extermination medal is that you don't have to kill the whole team, as long as they're all dead. ex: You get an Overkill in a 5v5 match, and the other player is on the respawn; as long as all players are on the respawn, and you receive an Overkill and the Extermination, whether you were the one who killed the whole team or not, which can be useful in Big Team battles. Inversely, you can also get the medal in a 2v2 match by killing both enemies twice, and spacing out the first 3 kills.
This is commonly received on Team SWAT, as you only have to get headshots, where it is very easy to obtain in Halo 3 on Snowbound due to the fact that all the opposing team members will be close to one another. You can also do it on Orbital by camping at the top of an escalator and waiting for a team of four to run up the stairs, you should have the upper hand because you have an ambush set, if you do it right and with a little luck you will get 4 headshots along with the medal. It is also more easily obtainable in Grifball, another easier way is to get an overkill in a 4v4 objective game, as the respawn time is longer, and if you aren't fast enough, you can spawn kill with a team mate as well.
It can also be obtained by killing the person who is in a team consisting only of himself, four times within four seconds of each kill.
In the event that you get an extermination medal on a Mythic Map it also unlocks the "Have Fun Respawning" achievement worth 25G.
In Halo: Reach, it is possible to earn an Extermination without getting an Overkill. The exact specifics are unknown, but the medal seems to require one to get 4 kills on the same team in about the maximum time you can get an overkill in (12 seconds) instead of the Overkill medal itself.