Java Edition:Wither Skeleton with Wither
Wither Skeletons inflicted with the wither effect were obtainable from 12w36a (1.4.2 snapshot) to 1.13.2. Both wither level I and level II are obtainable. The wither effect will eventually run out, making this a temporary discontinued feature.
Obtaining
From 12w36a (1.4.2 snapshot) to 1.13.2, wither skeletons were not immune to the wither effect. However, in 18w43a (1.14 snapshot) wither skeletons were made immune to the wither effect.[1] There are two methods to apply the wither effect, from the wither or from another wither skeleton.
10 Second Wither I
Wither skeletons will receive 10 seconds of the wither I effect if attacked by another wither skeleton. First, a wither skeleton must be made weaponless by the overstacked weapons or dual wielding methods. This weaponless wither skeleton must have CanPickUpLoot:1b so that it can pick up a bow. Next, have the wither skeleton wielding a bow shoot another wither skeleton. This will cause the shot wither skeleton to attack the bow wielding wither skeleton and give it the wither I effect for 10 seconds.
60 Second Wither I
From 12w34a to 12w34b, wither skulls gave Wither I for 60 seconds. Withers were unobtainable during this version range, and so the wither or wither skull must be downgraded from 12w36a or later into 12w34a - 12w34b. Wither skeletons were also added in 12w36a, after the version span of the long duration wither effect. However, a creeper could receive the 60 second wither I effect from 12w34a to 12w34b. If the creeper with 60 seconds of wither is updated to 16w32a - 1.13.2 and exploded, it will create an area effect cloud of 60 second wither I. Finally, a wither skeleton can enter this area effect cloud to gain the effect.
Wither II
Wither skeletons will receive 40 seconds of the wither II effect when hit by a skull launched by a wither on hard difficulty (or 10 seconds on normal difficulty). This works during the entire range of 12w36a to 1.13.2.