Many introverts are delegation-averse. Some are really, REALLY delegation-averse.
The issue has multiple facets:
Some introverts don’t want to deal with the “confrontational” issue of telling someone else what to do and then holding them accountable for doing it and doing it well.
Some introverts like the feeling of being self-reliant, and it feels like a loss of control to give tasks to others.
But most of all, many introverts do not want to have other people do the task because “they probably will do it wrong.” And “wrong” in this context, comes down to: “they won’t do it the way I do it.”
For introverts to grow as leaders, they need to make the conscious decision to build their competency in it. To do this effectively:
Start small, and delegate simple tasks first. Give bigger tasks as people prove their competency.
Focus on the output, not the process, since they may have a different method of arriving at the correct solution.
Show the respect of saying “thank you” when people complete the task.
Adopt a “first draft” mentality. Ask people to turn in a draft by an early deadline, and the introvert needs to assume that it is an imperfect DRAFT, not a final product. Consider this as an opportunity to grow the other person’s competency, so look it over and give feedback. Important: this feedback should be framed as “how to polish this to perfection,” NOT a red-pen markup of “this is WRONG, and THIS is REALLY WRONG.” Then communicate the polishing suggestions and give a new deadline for completion.
Consider that successful completion of a delegated task grows the other person’s competency, so the introvert can trust them to do it right the next time they are delegated a similar task.
Structure this to be an ongoing process. Consider adding something like “delegate 2 tasks” to daily to-do lists or work scheduling calendars.