From the Latin abrogātus, “repealed” (See ab-, rogation, -ate1).
The meaning has expanded from officially repealing a law (doing away with it) to delaying or avoiding to do something, like a chore or another responsibility.
EXAMPLES OF ABROGATE
By neglecting to attend the meetings, he was seen as trying to abrogate his duties.
The committee tried to abrogate its responsibility by postponing the decision until after the holidays.