What do you call a word that indicates the action is being done by, for, or to the antecedent?

Reflexive pronoun is the answer

The type of word that indicates the action is being done by, for, or to the antecedent is called a personal pronoun. Personal pronouns can be classified into two types: Subjective or Objective. Subjective personal pronouns are those pronouns that do the action. Objective personal pronouns are those pronouns that receive the action.