Purity of Revenge

Overview
The Silver Hand have attacked Jorrvaskr and Kodlak was killed in the fighting. Vilkas wants The Dragonborn to travel with him to attack the Silver Hand, retrieve the fragments of Wuuthrad, and avenge the death of Kodlak.
 * Travel with Vilkas to Driftshade Refuge, near Dawnstar.
 * Kill all members of the Silver Hand (optional).
 * Retrieve the shards of Wuuthrad.
 * Return to Jorrvaskr.

Revenge Served Cold
After the death of Kodlak Whitemane, Vilkas is obviously looking for revenge. With no questions asked, he will then act as a follower all the way to the Silver Hand hideout at Driftshade Refuge, a small fort southeast of Dawnstar. Note he will replace any existing follower, and it is not possible to share things with him, so he cannot be used to help carry things if the load becomes too large.

Upon arrival at the refuge, dispose of the three Silver Hand members guarding the exterior, then head inside.

Den of the Silver Hand
Not surprisingly, Driftshade Refuge is crawling with hostile Silver Hand members, so take advantage of Vilkas as a follower and kill the opposition one by one.

In the first room down the stairs, aside from Silver Hand members, there is an end table and regular table with loot on and in them. The door straight ahead (north) is barred. To the west is a room with several bookshelves, numerous books (most ruined or burned), and another table with random loot. To the north is another room with bookshelves and a table, and a pasage leads west to a wooden door.

Through that door is a two-leveled room with many boxes on the floor. Among the many books on bookshelves, there is An Explorer's Guide to Skyrim which will add a few map markers for later exploration. It is located on the upper level, on a bookshelf against the west wall beneath an elk head that's mounted on the wall. To the west on the lower level is an Adept-locked chest containing gold and potions. The tunnel north of the room leads to an Adept-locked door on the left, so unlock it and wipe out yet another Silver Hand, then loot the chest and the ingots from the table. In the next room, a set of spears blocks the entrance but the corresponding lever is easily located right next to the spears. Proceed forward and enter Driftshade Cellar.

On the left side behind barrels is a chest with leveled loot. Follow the path and make use of the Spike Wall trap to dispose of yet another enemy. Ignore the locked cell with the dead werewolf. Facing the Spike wall, activate it using the pull bar on the right and enter the area behind it for another treasure chest (taking care that Vilkas isn't in the way, doesn't matter as he is essential though). Hit a similar pullbar on the wall to get back out, and head east to door to another room with two enemies and, hidden on a shelf, the Light Armor skill book The Rear Guard. The cellar now turns into a snowy cave, and the next room is in two levels, with a Silver Hand archer on the above ledge. Kill him and examine the cages, one of them holds a living werewolf, who will attack you if you pick the lock; the other one contains a dead horse.

The Fragments of Wuuthrad
After passing through a torture chamber, the next large chamber contains only a couple enemies. A coin purse, two potions, and an Apothecary's Satchel filled with a few ingredients can be found on a bookshelf behind the fireplace. (The satchel is located on the top shelf of the bookcase, and can be easy to miss, as this is a darker corner of the room.)

Take the only available path, leading back to an initially inaccessible part of Driftshade Refuge (behind the barred door in the first room). The first chamber contains the three remaining members of the Silver Hand, the One-handed skill book 2920, Morning Star, v1, the Block skill book Warrior, a treasure chest with considerable loot and finally, the stolen Fragments of Wuuthrad. Use the door to the south for an easy escape route, then return to Jorrvaskr. As soon as the Dragonborn arrives, Vilkas will be quick to inform that the rest of the Companions should meet up at the Skyforge and pay their respects to the deceased, but now avenged, Whitemane. This will end the quest and start the next one, Glory of the Dead.

Trivia

 * In the Driftshade cellar, near the first dead werewolf encountered there is a trap with spikes. Behind the barrels there is a chest and a pullbar on the wall. If the pullbar is activated, the Dragonborn will be trapped. To get out, one may jump from the chest to the low set of barrels and then out (or find the other pullbar which opens the trap).
 * Note: if you have a follower with you, Vilkas will replace your follower. So unload important equipment given to your follower before approaching the companion's hall.
 * In dialogue with Vilkas, the Dragonborn has the option to ask "Did anyone else get hurt?" Vilkas will respond "I do not think so", however, Athis is clearly wounded and Ria tending to him.
 * If you're with Serana and she is replaced with Vilkas, you could simply talk to her to follow you, providing you 2 essential companions that cannot be killed. (Needs confirmation)

Bugs
Очистительная месть