Can you write a sequel to a book without being the original writer?

Answer 1

You may only write about another author's character if they give you legal permission to do so. Unless the original author is dead, you are probably not going to be hired to write a sequel to their work.

You may write fanfiction, which is writing that cannot be sold, but is just "passed around" on the internet for enjoyment. So long as you do not try to get any sort of "payment" or recognition, the author will probably ignore your writing.

Answer 2

You would need to get the author's permission to write a sequel to their story, if the author is still alive, as the story still belongs to the author.